a domain decomposition method based on weighted interior penalties for advection diffusion reaction problems we propose a domain decomposition method for advection diffusion reaction equations based on nitsche s transmission conditions the advection dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries we prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method the convergence of the resulting domain decomposition method is proved and this result holds true uniformly with respect to the diffusion parameter the numerical scheme that we propose here can thus be applied straightforwardly to diffusion dominated advection dominated and hyperbolic problems some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method and we compare our method with some other domain decomposition techniques for advection diffusion equations

a robust nitsche s formulation for interface problems abstract in this work we propose a novel weighting for the interfacial consistency terms arising in a nitsche variational form we demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method consequently we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of a elements with arbitrarily small volume fractions b large material heterogeneities and c both large heterogeneities as well as arbitrarily small elements we then highlight the accuracy and efficiency of the proposed formulation through numerical examples focusing particular attention on interfacial quantities of interest

mean variance optimization of discrete time discounted markov decision processes abstract in this paper we study a mean variance optimization problem in an infinite horizon discrete time discounted markov decision process mdp the objective is to minimize the variance of system rewards with the constraint of mean performance different from most of works in the literature which require the mean performance already achieve optimum we can let the discounted performance equal any constant the difficulty of this problem is caused by the quadratic form of the variance function which makes the variance minimization problem not a standard mdp by proving the decomposable structure of the feasible policy space we transform this constrained variance minimization problem to an equivalent unconstrained mdp under a new discounted criterion and a new reward function the difference of the variances of markov chains under any two feasible policies is quantified by a difference formula based on the variance difference formula a policy iteration algorithm is developed to find the optimal policy we also prove the optimality of deterministic policy over the randomized policy generated in the mean constrained policy space numerical experiments demonstrate the effectiveness of our approach

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions we develop a density based topology optimization method for linear elasticity based on the cut finite element method more precisely the design domain is discretized using cut finite elements whic

cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

geometrically accurate efficient and flexible quadrature techniques for the tetrahedral finite cell method abstract we illustrate the importance of geometrically accurate volume quadrature for obtaining optimal accuracy with non boundary fitted finite element discretizations when the problem domain is defined by sharp boundaries we consider the tetrahedral finite cell method tetfcm and replace its recursive subdivision based integration approach with geometrically accurate quadrature rules that emanate from higher order geometric parametrizations of cut tetrahedral elements the element wise parametrization procedure relies on the identification of the intersection topology and a series of higher order mappings based on lagrange polynomials we demonstrate with several 3d examples that geometrically faithful local parametrization ensures optimal accuracy while significantly reducing the number of quadrature points with respect to recursive subdivision on the other hand we highlight the strength of subdivision quadrature in the context of a patient specific workflow for the simulation based performance analysis of coupled bone implant configurations in particular we show that accuracy flexibility and computational efficiency of the tetfcm critically depend on flexibly applying the two different quadrature variants for fuzzy imaging data and sharp boundary representations respectively

optimization of markov decision processes under the variance criterion in this paper we study a variance minimization problem in an infinite stage discrete time markov decision process mdp regardless of the mean performance for the markov chain under the variance criterion since the value of the cost function at the current stage will be affected by future actions this problem is not a standard mdp and the traditional mdp theory is not applicable in this paper we convert the variance minimization problem into a standard mdp by introducing a concept called pseudo variance then we derive a variance difference formula that quantifies the difference of variances of markov systems under any two policies with the difference formula the correlation of the variance cost function at different stages can be decoupled through a nonnegative term a necessary condition of the optimal policy is obtained it is also proved that the optimal policy with the minimal variance can be found in the deterministic policy space furthermore we propose an efficient iterative algorithm to reduce the variance of markov systems we prove that this algorithm can converge to a local optimum finally a numerical experiment is conducted to demonstrate the efficiency of our algorithm compared with the gradient based method widely adopted in the literature

cut finite element methods for elliptic problems on multipatch parametric surfaces we develop a finite element method for the laplace beltrami operator on a surface described by a set of patchwise parametrizations the patches provide a partition of the surface and each patch is

a hybrid finite volume finite element method for bulk surface coupled problems abstract the paper develops a hybrid method for solving a system of advection diffusion equations in a bulk domain coupled to advection diffusion equations on an embedded surface a monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface in our approach the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way moreover a triangulation of the surface into regular shaped elements is not required the background mesh is an octree grid with cubic cells as an example of an application we consider the modeling of contaminant transport in fractured porous media one standard model leads to a coupled system of advection diffusion equations in a bulk matrix and along a surface fracture a series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach the method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures

a stabilised immersed boundary method on hierarchical b spline grids for fluid rigid body interaction with solid solid contact abstract an accurate efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly supported rigid bodies and incompressible fluid flow with topology changes and solid solid contact the solution of the incompressible navier stokes equations is approximated by employing a stabilised formulation on cartesian grids discretised with hierarchical b splines the solid is modelled as a rigid body and represented by linear segments along its boundary kinematic conditions along the fluid rigid body interface are enforced weakly using nitsche s method while ghost penalty operators are employed to avoid excessive ill conditioning of the system matrix arising from small cut cells a staggered scheme is used for resolving the coupled fluid rigid body interaction the contact between moving or moving and fixed solid bodies is modelled with lagrange multipliers the excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems the examples cover the galloping phenomena particulate flow hydraulic check valves and a model turbine

a nurbs based generalized finite element scheme for 3d simulation of heterogeneous materials a 3d nurbs based interface enriched generalized finite element method nigfem is introduced to solve problems with complex discontinuous gradient fields observed in the analysis of heterogeneous materials the method utilizes simple structured meshes of hexahedral elements that do not necessarily conform to the material interfaces in heterogeneous materials by avoiding the creation of conforming meshes used in conventional fem the nigfem leads to significant simplification of the mesh generation process to achieve an accurate solution in elements that are crossed by material interfaces the nigfem utilizes non uniform rational b splines nurbs to enrich the solution field locally the accuracy and convergence of the nigfem are tested by solving a benchmark problem we observe that the nigfem preserves an optimal rate of convergence and provides additional advantages including the accurate capture of the solution fields in the vicinity of material interfaces and the built in capability for hierarchical mesh refinement finally the use of the nigfem in the computational analysis of heterogeneous materials is discussed development of generalized finite element method for 3d mesoscale modeling of materials with complex microstructures novel nurbs based enrichment functions built in capability to perform hierarchical local mesh refinement method is applicable to engineering problems characterized by discontinuous gradient and complex internal geometry

cut finite element methods for coupled bulk surface problems we develop a cut finite element method for a second order elliptic coupled bulk surface model problem we prove a priori estimates for the energy and l 2 l2 norms of the error using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method finally we present a numerical example illustrating the accuracy and the robustness of our approach

a cut finite element method for a model of pressure in fractured media we develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks the crack has its own pressure field and can cut through

numerical integration for nonlinear problems of the finite cell method using an adaptive scheme based on moment fitting abstract fictitious domain methods such as the finite cell method simplify the discretization process significantly as the mesh is decoupled from the geometrical description however this simplification in the mesh generation results in broken cells which is why special integration methods are required usually adaptive integration schemes are applied resulting in a large number of integration points and thus an expensive numerical integration especially for nonlinear applications to perform the numerical integration more efficiently we propose an adaptive integration method using moment fitting thereby we present a moment fitting approach based on lagrange polynomials through gauss legendre points to circumvent having to solve the moment fitting equation system the performance of this integration method is shown by studying several numerical examples of the finite cell method for small and large strain problems in elastoplasticity

shape optimization using the cut finite element method we present a cut finite element method for shape optimization in the case of linear elasticity the elastic domain is defined by a level set function and the evolution of the domain is obtained by

a discontinuous galerkin based immersed boundary method with non homogeneous boundary conditions and its application to elasticity we propose a discontinuous galerkin based immersed boundary method for elasticity problems the resulting numerical scheme does not require boundary fitting meshes and avoids boundary locking by switching the elements intersected by the boundary to a discontinuous galerkin approximation special emphasis is placed on the construction of a method that retains an optimal convergence rate in the presence of non homogeneous essential and natural boundary conditions the role of each one of the approximations introduced is illustrated by analyzing an analog problem in one spatial dimension finally extensive two and three dimensional numerical experiments on linear and nonlinear elasticity problems verify that the proposed method leads to optimal convergence rates under combinations of essential and natural boundary conditions

an unfitted nitsche method for incompressible fluid structure interaction using overlapping meshes we consider the extension of the nitsche method to the case of fluid structure interaction problems on unfitted meshes we give a stability analysis for the space semi discretized problem and show how this estimate may be used to derive optimal error estimates for smooth solutions irrespectively of the mesh interface intersection we also discuss different strategies for the time discretization using either fully implicit or explicit coupling loosely coupled schemes some numerical examples illustrate the theoretical discussion

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

a new least squares stabilized nitsche method for cut isogeometric analysis we derive a new stabilized symmetric nitsche method for enforcement of dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis cutiga we consider c1 spli

markov decision processes with a minimum variance criterion abstract we consider the optimization of the variance of the sum of costs as well as that of an average expected cost in markov decision processes with unbounded cost in case of general state and action space we find the stationary policy which makes the average variance as small as possible in the class of policies which are e optimal in an average expected cost

the diffuse nitsche method dirichlet constraints on phase field boundaries summary we explore diffuse formulations of nitsche s method for consistently imposing dirichlet boundary conditions on phase field approximations of sharp domains leveraging the properties of the phase field gradient we derive the variational formulation of the diffuse nitsche method by transferring all integrals associated with the dirichlet boundary from a geometrically sharp surface format in the standard nitsche method to a geometrically diffuse volumetric format we also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem from which the stabilization parameter can be estimated automatically in each element we advertise metastable phase field solutions of the allen cahn problem for transferring complex imaging data into diffuse geometric models in particular we discuss the use of mixed meshes that is an adaptively refined mesh for the phase field in the diffuse boundary region and a uniform mesh for the representation of the physics based solution fields we illustrate accuracy and convergence properties of the diffuse nitsche method and demonstrate its advantages over diffuse penalty type methods in the context of imaging based analysis we show that the diffuse nitsche method achieves the same accuracy as the standard nitsche method with sharp surfaces if the inherent length scales i e the interface width of the phase field the voxel spacing and the mesh size are properly related we demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body this article is protected by copyright all rights reserved

optimal control of ergodic continuous time markov chains with average sample path rewards in this paper we study continuous time markov decision processes with the average sample path reward aspr criterion and possibly unbounded transition and reward rates we propose conditions on the system s primitive data for the existence of epsilon aspr optimal deterministic stationary policies in a class of randomized markov policies satisfying some additional continuity assumptions the proof of this fact is based on the time discretization technique the martingale stability theory and the concept of potential we also provide both policy and value iteration algorithms for computing or at least approximating the epsilon aspr optimal stationary policies we illustrate with examples our main results as well as the difference between the aspr and the average expected reward criteria

a combined finite element method for elliptic problems posted in domains with rough boundaries abstract a combined finite element method is presented in this paper to solve the elliptic problems posted in domains with rough boundaries solving these problems numerically is difficult because resolving the boundaries usually requires very fine meshes while good quality meshes often over refine unnecessarily the interior of the domain the basic idea of the proposed method is to use a fine mesh with size h in the vicinity of oscillating boundaries and a coarse mesh with size h h for other portions of the domain to reduce some unnecessary computational effort the transmission conditions across the fine coarse mesh interface are treated by the penalty technique the key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form which avoids the affection of the ratio h h in the error estimate we prove a quasi optimal convergence in terms of elements since there is no whole h 2 regularity in the domain with rough boundaries numerical results are provided for elliptic equations in domains with non oscillating or oscillating boundaries to illustrate the theoretical results

isogeometric analysis on v reps first results abstract inspired by the introduction of volumetric modeling via volumetric representations v reps by massarwi and elber in 2016 in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic pdes on trimmed geometries seen as a special class of more general v reps we develop tools for approximation and local re parameterization of trimmed elements for three dimensional problems and we provide a theoretical framework that fully justifies our algorithmic choices we validate our approach both on two and three dimensional problems for diffusion and linear elasticity

a new face oriented stabilized xfem approach for 2d and 3d incompressible navier stokes equations abstract we develop a new stabilized xfem based fixed grid approach for the transient incompressible navier stokes equations using cut elements our framework is based on a new flexible strategy to handle several degrees of freedom dofs at nodes required for complex shaped embedded meshes as they occur for fixed grid fluid structure interaction problems boundary conditions on embedded boundaries are imposed weakly using a nitsche type approach ghost penalty terms for velocity and pressure are added for stability reasons and to improve the conditioning of the system matrix the idea of ghost penalties previously developed for stokes problems is extended to the incompressible navier stokes equations by using face oriented fluid stabilizations for both viscous and convective dominated flows furthermore the need for additional terms to enforce boundary conditions for non viscous flows is addressed a detailed numerical study of our stabilized fluid formulation including studies of nitsche s parameter and the ghost penalty parameter shows optimal error convergence and a good system conditioning in the viscous and the convective dominated cases compared to other fixed grid fluid formulations our method is much more accurate and less sensitive to the location of the interface regarding the accuracy and the sensitivity with respect to the interface position a clear improvement could be observed for the viscous and pressure fluxes as well as for the imposition of the boundary condition results of a transient two dimensional high reynolds number flow over a cylinder and the flow over a thick plate in 3d confirm the applicability of the proposed method

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

a new fictitious domain approach inspired by the extended finite element method the purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by moes dolbow and belytschko in internat j numer methods engrg 46 1999 pp 131 150 an optimal method is obtained thanks to an additional stabilization technique some a priori estimates are established and numerical experiments illustrate different aspects of the method the presentation is made on a simple poisson problem with mixed neumann and dirichlet boundary conditions the extension to other problems or boundary conditions is quite straightforward

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

cut finite element modeling of linear membranes we construct a cut finite element method for the membrane elasticity problem on an embedded mesh using tangential differential calculus i e with the equilibrium equations pointwise projected ont

zero sum stochastic games in borel spaces average payoff criteria this paper is concerned with two person zero sum dynamic stochastic games in borel spaces with possibly unbounded payoff function and several average or ergodic payoff criteria we give conditions under which the long run expected average payoff criterion the sample path average criterion the existence of solutions to the average payoff shapley equations and a certain martingale condition are all equivalent

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

another set of conditions for markov decision processes with average sample path costs abstract this paper deals with discrete time markov decision processes with average sample path costs aspc in borel spaces the costs may have neither upper nor lower bounds we propose new conditions for the existence of e aspc optimal deterministic stationary policies in the class of all randomized history dependent policies our conditions are weaker than those in the previous literature moreover some sufficient conditions for the existence of aspc optimal stationary policies are imposed on the primitive data of the model in particular the stochastic monotonicity condition in this paper has first been used to study the aspc criterion also the approach provided here is slightly different from the optimality equation approach widely used in the previous literature on the other hand under mild assumptions we show that average expected cost optimality and aspc optimality are equivalent finally we use a controlled queueing system to illustrate our results

graded parametric cutfem and cutiga for elliptic boundary value problems in domains with corners we develop a parametric cut finite element method for elliptic boundary value problems with corner singularities where we have weighted control of higher order derivatives of the solution to a neig

an implicit boundary finite element method with extension to frictional sliding boundary conditions and elasto plastic analyses abstract implicit boundary methods which enrich the interpolation structure with implicit weight functions are straightforward methods for the enforcement of dirichlet boundary conditions in this article we follow the implicit boundary method that uses approximate step functions the step boundary method developed by kumar et al and provide modifications that have several advantages roller boundary conditions have wide practical applications in engineering however the step boundary method for roller boundary conditions with inclinations has yet to be fully formulated through to the final linear system of equations thus we provide a complete derivation that leads to simplified stiffness matrices compared to the original approach which can be implemented directly in fictitious domain finite element analysis the approach is then extended we believe for the first time to the nonlinear cases of frictional boundary conditions and elasto plastic material behaviour the proposed formulation and procedures are validated on a number of example problems that test different aspects of the method

a nitsche cut finite element method for the oseen problem with general navier boundary conditions abstract in this work a nitsche based imposition of generalized navier conditions on cut meshes for the oseen problem is presented other methods from literature dealing with the generalized navier condition impose this condition by means of substituting the tangential robin condition in a classical galerkin way these methods work fine for a large slip length coefficient but lead to conditioning and stability issues when it approaches zero we introduce a novel method for the weak imposition of the generalized navier condition which remains well posed and stable for arbitrary choice of slip length including zero the method proposed here builds on the formulation done by juntunen and stenberg 2009 they impose a robin condition for the poisson problem by means of nitsche s method for an arbitrary combination of the dirichlet and neumann parts of the condition the analysis conducted for the proposed method is done in a similar fashion as in massing et al 2018 but is done here for a more general type of boundary condition the analysis proves stability for all flow regimes and all choices of slip lengths also an l 2 optimal estimate for the velocity error is shown which was not conducted in the previously mentioned work a numerical example is carried out for varying slip lengths to verify the robustness and stability of the method with respect to the choice of slip length even though proofs and formulations are presented for the more general case of an unfitted grid method they can easily be reduced to the simpler case of a boundary fitted grid with the removal of the ghost penalty stabilization terms

fictitious domain finite element methods using cut elements i a stabilized lagrange multiplier method we propose a fictitious domain method where the mesh is cut by the boundary the primal solution is computed only up to the boundary the solution itself is defined also by nodes outside the domain

some new results on sample path optimality in ergodic control of diffusions we present some new results on sample path optimality for the ergodic control problem of a class of nondegenerate diffusions controlled through the drift the hypothesis most often used in the literature to ensure the existence of an almost sure sample path optimal stationary markov control requires finite second moments of the first hitting times tau of bounded domains over all admissible controls we show that this can be considerably weakened mathbb e tau 2 may be replaced with mathbb e tau ln tau thus reducing the required rate of convergence of averages from polynomial to logarithmic a foster lyapunov condition that guarantees this is also exhibited moreover we study a large class of models that are neither uniformly stable nor have a near monotone running cost and we exhibit sufficient conditions for the existence of a sample path optimal stationary markov control

imposing dirichlet boundary conditions in hierarchical cartesian meshes by means of stabilized lagrange multipliers this is the pre peer reviewed version of the following article tur m albelda j nadal e and rodenas j j 2014 imposing dirichlet boundary conditions in hierarchical cartesian meshes by means of stabilized lagrange multipliers int j numer meth engng 98 399 417 which has been published in final form at http dx doi org 10 1002 nme 4629 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

asymptotic analysis of model problems for a coupled system on etudie le comportement asymptotique quand e 0 des solutions du systeme d equations couplees u e f u u 0 pour f u u u ou u qui satisfait les conditions de dirichlet u 0 e u 1 e 0 et 0 e 0 1 e 1

bias optimality and strong n n 1 0 discount optimality for markov decision processes abstract in this paper we study both bias optimality and strong n n 1 0 discount optimality for denumerable discrete time markov decision processes the rewards may have neither upper nor lower bounds we give sufficient conditions on the system s primitive data and under which we prove 1 the existence of the bias optimality equation and bias optimal policies 2 a condition equivalent to bias optimal policies 3 average expected reward optimality and strong 1 discount optimality are equivalent 4 bias optimality and strong 0 discount optimality are equivalent 5 the existence of strong n n 1 0 discount optimal stationary policies our conditions are weaker than those in the previous literature moreover our results are illustrated by a controlled random walk

recurrence conditions for average and blackwell optimality in denumerable state markov decision chains in a previous paper dekker and hordijk 1988 presented an operator theoretical approach for multichain markov decision processes with a countable state space compact action sets and unbounded rewards conditions were presented guaranteeing the existence of a laurent series expansion for the discounted rewards the existence of average and blackwell optimal policies and the existence of solutions for the average and blackwell optimality equations while these assumptions were operator oriented and formulated as conditions for the deviation matrix we will show in this paper that the same approach can also be carried out under recurrence conditions these new conditions seem easier to check in general and are especially suited for applications in queueing models

zero sum semi markov games this paper deals with borel state and action spaces zero sum semi markov games under the expected long run average payoff criterion the transition probabilities are assumed to satisfy some generalized geometric ergodicity conditions the main result states that the optimality equation has a solution which is approximated by the solutions of some varepsilon perturbed semi markov games as a corollary the existence of value and average optimal strategies for the players is established

a mean variance optimization problem for discounted markov decision processes in this paper we consider a mean variance optimization problem for markov decision processes mdps over the set of deterministic stationary policies different from the usual formulation in mdps we aim to obtain the mean variance optimal policy that minimizes the variance over a set of all policies with a given expected reward for continuous time mdps with the discounted criterion and finite state and action spaces we prove that the mean variance optimization problem can be transformed to an equivalent discounted optimization problem using the conditional expectation and markov properties then we show that a mean variance optimal policy and the efficient frontier can be obtained by policy iteration methods with a finite number of iterations we also address related issues such as a mutual fund theorem and illustrate our results with an example

a stabilized nitsche overlapping mesh method for the stokes problem we develop a nitsche based formulation for a general class of stabilized finite element methods for the stokes problem posed on a pair of overlapping non matching meshes by extending the least squares stabilization to the overlap region we prove that the method is stable consistent and optimally convergent to avoid an ill conditioned linear algebra system the scheme is augmented by a least squares term measuring the discontinuity of the solution in the overlap region of the two meshes as a consequence we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface finally we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results

sample path optimality and variance minimization of average cost markov control processes this paper studies several average cost criteria for markov control processes on borel spaces with possibly unbounded costs under suitable hypotheses we show i the existence of a sample path average cost spac optimal stationary policy ii a stationary policy is spac optimal if and only if it is expected average cost eac optimal and iii within the class of stationary spac optimal equivalently eac optimal policies there exists one with a minimal limiting average variance

nitsche s method for general boundary conditions we introduce a method for treating general boundary conditions in the finite element method generalizing an approach due to nitsche 1971 for approximating dirichlet boundary conditions we use poisson s equations as a model problem and prove a priori and a posteriori error estimates the method is also compared with the traditional galerkin method the theoretical results are verified numerically

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

a nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity in this note we propose a finite element method for incompressible or compressible elasticity problems with discontinuous modulus of elasticity or if compressible poisson s ratio the problem is written on mixed form using p1 continuous displacements and elementwise p0 pressures leading to the possibility of eliminating the pressure beforehand in the compressible case in the incompressible case the method is augmented by a stabilization term penalizing the pressure jumps we show a priori error estimates under certain regularity hypothesis in particular we prove that if the exact solution is sufficiently smooth in each subdomain then the convergence order is optimal

optimal convergence analysis for the extended finite element method we establish some optimal a priori error estimate on some variants of the extended finite element method xfem namely the xfem with a cut off function and the standard xfem with a fixed enrichment area the results are established for the lame system homogeneous isotropic elasticity and the laplace problem the convergence of the numerical stress intensity factors is also investigated we show some numerical experiments which corroborate the theoretical results

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

reissner mindlin shell theory based on tangential differential calculus abstract the linear reissner mindlin shell theory is reformulated in the frame of the tangential differential calculus tdc using a global cartesian coordinate system the rotation of the normal vector is modelled with a difference vector approach the resulting equations are applicable to both explicitly and implicitly defined shells because the employed surface operators do not necessarily rely on a parametrization hence shell analysis on surfaces implied by level set functions is enabled but also the classical case of parametrized surfaces is captured as a consequence the proposed tdc based formulation is more general and may also be used in recent finite element approaches such as the tracefem and cutfem where a parametrization of the middle surface is not required herein the numerical results are obtained by isogeometric analysis using nurbs as trial and test functions for classical and new benchmark tests in the residual errors optimal higher order convergence rates are confirmed when the involved physical fields are sufficiently smooth

a stabilized nitsche cut finite element method for the oseen problem abstract we provide the numerical analysis for a nitsche based cut finite element formulation for the oseen problem which has been originally presented for the incompressible navier stokes equations by schott and wall 2014 and allows the boundary of the domain to cut through the elements of an easy to generate background mesh the formulation is based on the continuous interior penalty cip method of burman et al 2006 which penalizes jumps of velocity and pressure gradients over inter element faces to counteract instabilities arising for high local reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure since the mesh does not fit the boundary dirichlet boundary conditions are imposed weakly by a stabilized nitsche type approach the addition of cip like ghost penalties in the boundary zone allows to prove that our method is inf supstable and to derive optimal order a priori error estimates in an energy type norm irrespective of how the boundary cuts the underlying mesh all applied stabilization techniques are developed with particular emphasis on low and high reynolds numbers two and three dimensional numerical examples corroborate the theoretical findings finally the proposed method is applied to solve the transient incompressible navier stokes equations on a complex geometry

a note on the stability parameter in nitsche s method for unfitted boundary value problems abstract nitsche s method is a popular approach to implement dirichlet type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible the method is widely applied in the context of unfitted finite element methods of the classical symmetric nitsche s method it is well known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems in this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this we show that in specific situations error bounds can deteriorate and give examples of computations where nitsche s method yields large and even diverging discretization errors

nitsche xfem for the coupling of an incompressible fluid with immersed thin walled structures abstract in this paper we introduce a nitsche xfem method for fluid structure interaction problems involving a thin walled elastic structure lagrangian formalism immersed in an incompressible viscous fluid eulerian formalism the fluid domain is discretized with an unstructured mesh not fitted to the solid mid surface mesh weak and strong discontinuities across the interface are allowed for the velocity and pressure respectively the fluid solid coupling is enforced consistently using a variant of nitsche s method with cut elements robustness with respect to arbitrary interface intersections is guaranteed through suitable stabilization several coupling schemes with different degrees of fluid solid time splitting implicit semi implicit and explicit are investigated a series of numerical test in 2d involving static and moving interfaces illustrates the performance of the different methods proposed

analysis of the shifted boundary method for the stokes problem abstract the analysis of stability and accuracy of the shifted boundary method is developed for the stokes flow equations the key feature of the shifted boundary method an embedded finite element method is the shifting of the location where boundary conditions are applied from the true to a surrogate boundary and an appropriate modification shifting of the value of the boundary conditions an inf sup condition is proved for the variational formulation associated to the shifted boundary method and we derive by way of strang s second lemma an optimal error estimate in the natural sbm norm we also derive an l 2 error estimate for the velocity field by means of an extension of the aubin nitsche approach to embedded non consistent mixed finite element methods

higher order conformal decomposition fem cdfem abstract a higher order accurate finite element method is proposed which uses automatically generated meshes based on implicit level set data for the description of boundaries and interfaces in two and three dimensions the method is an alternative for fictitious domain and extended finite element methods the domain of interest is immersed in a background mesh composed by higher order elements the zero level sets are identified and meshed followed by a decomposition of the cut background elements into conforming sub elements adaptivity is a crucial ingredient of the method to guarantee the success of the mesh generation it ensures the successful decomposition of cut elements and enables improved geometry descriptions and approximations it is confirmed that higher order accurate results with optimal convergence rates are achieved with the proposed conformal decomposition finite element method cdfem

a stabilized cut finite element method for the three field stokes problem we propose a nitsche based fictitious domain method for the three field stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh the dependent variables of velocity pressure and extra stress tensor are discretized on the background mesh using linear finite elements this equal order approximation is stabilized using a continuous interior penalty cip method on the unfitted domain boundary dirichlet boundary conditions are weakly enforced using nitsche s method we add cip like ghost penalties in the boundary region and prove that our scheme is inf sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh additionally we demonstrate that the condition number of the system matrix is bounded independently of the boundary location we corroborate our theoretical findings numerically

an immersed weak galerkin method for elliptic interface problems abstract in this paper we present an immersed weak galerkin method for solving second order elliptic interface problems the proposed method does not require solution meshes to be aligned with the interface consequently uniform cartesian meshes can be used for nontrivial interfacial geometry we show the existence and uniqueness of the numerical algorithm and provide the error analysis in the energy norm numerical results are reported to demonstrate the performance of the method

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

solutions of the average cost optimality equation for markov decision processes with weakly continuous kernel the fixed point approach revisited abstract this paper shows the existence of lower semicontinuous solutions of the average cost optimality equation for markov decision processes with borel spaces possible unbounded cost function and weakly continuous transition kernel this is done imposing a growth condition on the cost function a lyapunov stability condition on the transition kernel and a set of standard compactness continuity conditions the solution of the average cost optimality equation is obtained by means of the banach fixed point theorem

efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions in recent years a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on nonmatching or overlapping meshes examples of such methods are the fictitious domain method the extended finite element method and nitsche s method in all these methods integrals must be computed over cut cells or subsimplices which is challenging to implement especially in three space dimensions in this note we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices as a canonical example of an overlapping mesh method we consider nitsche s method which we apply to poisson s equation and a linear elastic problem

high degree discontinuous petrov galerkin immersed finite element methods using fictitious elements for elliptic interface problems abstract we propose a new strategy for constructing the p th degree immersed finite element ife spaces by applying the least squares framework in adjerid et al 2017 on fictitious elements this new construction method significantly reduces the ill conditioning caused by the small subelement issue in adjerid et al 2017 of solving the local ife shape functions the proposed ife spaces are employed in a discontinuous petrov galerkin dpg scheme to solve the second order elliptic interface problems we present a group of numerical examples to show that the dpgife method with the new p th degree ife space as the trial function space has the optimal convergence rate which improves the numerical results reported in adjerid et al 2017

refined approximations of the solutions of a coupled system with turning points abstract we present in this paper the asymptotic behavior of solutions of a boundary value problem for a coupled system of differential equations u e f u u g x u u 0 on a compact interval i where f u u has turning points in i we provide upper and lower solutions x e and x e respectively which bound solutions exhibiting boundary layer and interior layer behavior for which lime 0 x e x e 0 uniformly on i

zero sum average semi markov games fixed point solutions of the shapley equation this paper deals with zero sum average semi markov games with borel state and action spaces unbounded payoffs and mean holding times a solution to the shapley equation is obtained via the banach fixed point theorem assuming that the model satisfies a lyapunov like condition a growth hypothesis on the payoff function and the mean holding times besides standard continuity and compactness requirements

a nitsche based cut finite element method for the coupling of incompressible fluid flow with poroelasticity abstract the focus of this contribution is the numerical treatment of interface coupled problems concerning the interaction of incompressible fluid flow and permeable elastic structures the main emphasis is on extending the range of applicability of available formulations on especially three aspects these aspects are the incorporation of a more general poroelasticity formulation the use of the cut finite element method cutfem to allow for large interface motion and topological changes of the fluid domain and the application of a novel nitsche based approach to incorporate the beavers joseph saffmann tangential interface condition this last aspect allows one to extend the practicable range of applicability of the proposed formulation down to very low porosities and permeabilities which is important in several examples in application different aspects of the presented formulation are analyzed in a numerical example including spatial convergence the sensitivity of the solution to the nitsche penalty parameters varying porosities and permeabilities and a varying beavers joseph interface model constant finally a numerical example analyzing the fluid induced bending of a poroelastic beam provides evidence of the general applicability of the presented approach

sample path average optimality for markov control processes the authors consider a markov control process with borel state and actions spaces unbounded costs and under the long run sample path average cost criterion they prove that under very weak assumptions on the transition law and a moment assumption for the one step cost there exists a stationary policy with invariant probability distribution v that is sample path average cost optimal for v almost all initial states in addition every expected average cost optimal stationary policy is in fact liminf sample path average cost optimal and strongly expected average cost optimal

projection based reduced order models for a cut finite element method in parametrized domains abstract this work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method such methods and in particular the cutfem method are attractive in the generation of projection based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes the combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems avoiding remeshing as well as the reference domain formulation often used in the reduced order modeling for boundary fitted finite element formulations the resulting novel methodology is presented on linear elliptic and stokes problems together with several test cases to assess its capability the role of a proper extension and transport of embedded solutions to a common background is analyzed in detail

finite element approximation of the laplace beltrami operator on a surface with boundary we develop a finite element method for the laplace beltrami operator on a surface with boundary and nonhomogeneous dirichlet boundary conditions the method is based on a triangulation of the surfa

a high order discontinuous galerkin method for compressible flows with immersed boundaries summary we present a higher order discretization scheme for the compressible euler and navier stokes equations with immersed boundaries our approach makes use of a discontinuous galerkin dg discretization in a domain that is implicitly defined by means of a level set function the zero iso contour of this level set function is considered as an additional domain boundary where we weakly enforce boundary conditions in the same manner as in boundary fitted cells in order to retain the full order of convergence of the scheme it is crucial to perform volume and surface integrals in boundary cells with high accuracy this is achieved using a linear moment fitting strategy moreover we apply a non intrusive cell agglomeration technique that averts problems with very small and ill shaped cuts the robustness accuracy and convergence properties of the scheme are assessed in several two dimensional test cases for the steady compressible euler and navier stokes equations approximation orders range from zero to four even though the approach directly generalizes to even higher orders in all test cases with a sufficiently smooth solution the experimental order of convergence matches the expected rate for dg schemes this article is protected by copyright all rights reserved

approximate static condensation algorithm for solving multi material diffusion problems on meshes non aligned with material interfaces abstract in this paper we describe a new algorithm for solving multi material diffusion problem when material interfaces are not aligned with the mesh in this case interface reconstruction methods are used to construct approximate representation of interfaces between materials they produce so called multi material cells in which materials are represented by material polygons that contain only one material the reconstructed interface is not continuous between cells we suggest the new method for solving multi material diffusion problems on such meshes and compare its performance with known homogenization methods

a high order discontinuous galerkin nitsche method for elliptic problems with fictitious boundary we present a discontinuous galerkin method based on the classical method of nitsche for elliptic problems with an immersed boundary representation on a structured grid in such methods very small elements typically occur at the boundary leading to breakdown of the discrete coercivity as well as numerical instabilities in this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain this construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates furthermore we prove a bound of the condition number of the stiffness matrix all the results are valid for polynomials of arbitrary order we also discuss the implementation of the method and present numerical examples in three dimensions

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality

multiscale methods for problems with complex geometry we propose a multiscale method for elliptic problems on complex domains e g domains with cracks or complicated boundary for local singularities this paper also offers a discrete alternative to enrichment techniques such as xfem we construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account the corrections only need to be computed in regions surrounding fine scale geometric features we achieve linear convergence rate in the energy norm for the multiscale solution moreover the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh the analytical findings are verified in a series of numerical experiments

condition number analysis and preconditioning of the finite cell method the isogeometric finite cell method in which a domain is immersed in a structured background mesh suffers from conditioning problems when cells with small volume fractions occur in this contribution we establish a rigorous scaling relation between the condition number of i fcm system matrices and the smallest cell volume fraction ill conditioning stems either from basis functions being small on cells with small volume fractions or from basis functions being nearly linearly dependent on such cells based on these two sources of ill conditioning an algebraic preconditioning technique is developed which is referred to as symmetric incomplete permuted inverse cholesky sipic a detailed numerical investigation of the effectivity of the sipic preconditioner in improving i fcm condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the poisson problem and for two and three dimensional problems in linear elasticity in which nitche s method is applied in either the normal or tangential direction the accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method