automated code generation for discontinuous galerkin methods a compiler approach for generating low level computer code from high level input for discontinuous galerkin finite element forms is presented the input language mirrors conventional mathematical notation and the compiler generates efficient code in a standard programming language this facilitates the rapid generation of efficient code for general equations in varying spatial dimensions key concepts underlying the compiler approach and the automated generation of computer code are elaborated the approach is demonstrated for a range of common problems including the poisson biharmonic advection diffusion and stokes equations

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

fast algorithms for intersection of non matching grids using plcker coordinates the xfem and mortar methods can be used in combination with non matching or non conforming grids to deal with problems on complex geometries however the information about the mesh intersection must be provided we present algorithms for intersections between 1d and 2d unstructured multi component simplicial meshes and their intersections with a background unstructured 3d mesh a common algorithm based on the advancing front technique is used for the efficient selection of candidate pairs among simplicial elements bounding interval hierarchy bih of axes aligned bounding boxes aabb of elements is used to initialize the front tracking algorithm the family of element intersection algorithms is built upon a linetriangle intersection algorithm based on the plcker coordinates these algorithms combined with the advancing front technique can reuse the results of calculations performed on the neighboring elements and reduce the number of arithmetic operations barycentric coordinates on each of the intersecting elements are provided for every intersection point benchmarks of the element intersection algorithms are presented and three variants of the global intersection algorithm are compared on the meshes raising from hydrogeological applications

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

on asymptotic properties of maximum likelihood estimators for parabolic stochastic pde s we investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic spdes of the form du t x a 0 theta a 1 u t x dt dw t x wherea 0 anda 1 are partial differential operators andw is a cylindrical brownian motion we introduce a spectral method for computing mles based on finite dimensional approximations to solutions of such systems and establish criteria for consistency asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity we derive the asymptotic properties of the mle from a condition on the order of the operators in particular the mle is consistent if and only if ord a 1 1 2 ord a 0 a 1 d

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

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

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

on the aleksandrov bakel man pucci estimate for some elliptic and parabolic nonlinear operators in this work we prove the aleksandrov bakel man pucci estimate for possibly degenerate nonlinear elliptic and parabolic equations of the form rm div left f left nabla u x right right f left x right quad rm in omega subset mathbb r n and u t x t rm div left f left nabla u x t right right f left x t right quad rm in q subset mathbb r n 1 for f a fancyscript c 1 monotone field under some suitable conditions examples of applications such as the p laplacian and the mean curvature flow are considered as well as extensions of the general results to equations that are not in divergence form such as the m curvature flow

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

immersed finite element method abstract in this paper the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models in ifem a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain hence the mesh generation is greatly simplified moreover both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub domains are enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method rkpm delta function in comparison with the immersed boundary ib method the higher ordered rkpm delta function enables the fluid domain to have nonuniform spatial meshes with arbitrary geometries and boundary conditions the use of such kernel functions may eventually open doors to multi scale and multi resolution modelings of complex fluid structure interaction problems rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method the results compare well with the experimental data to the authors knowledge these are the first solutions that deal with particulate flows with very flexible solids

a comment on the article a finite element method for simulation of strong and weak discontinuities in solid mechanics by a hansbo and p hansbo comput methods appl mech engrg 193 2004 3523 3540

an alexandroff bakelman pucci estimate on riemannian manifolds abstract in cabre 1997 2 cabre established an alexandroff bakelman pucci abp estimate on riemannian manifolds with non negative sectional curvatures and applied it to establish the krylov safonov harnack inequality on manifolds with non negative sectional curvatures in the present paper we generalize the results of 2 we obtain an abp estimate on manifolds with ricci curvatures bounded from below and apply this estimate to prove the krylov safonov harnack inequality on manifolds with sectional curvatures bounded from below we also use this abp estimate to study minkowski type inequalities

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

unfitted extended finite elements for composite grids abstract we consider an extended finite elements method to handle the case of composite independent unstructured grids that lead to unfitted meshes in particular we address the case of two overlapped meshes a background and a foreground one where the thickness of the latter is smaller than the elements of the background mesh this situation may lead to elements split into several portions thus generating polyhedral elements we detail the corresponding discrete formulation for the poisson problem with discontinuous coefficients we also provide some technical details for the 3d implementation finally we provide some numerical examples with the aim of showing the effectiveness of the proposed formulation

a dlm fd method for fluid flexible body interactions in this study we extended the distributed lagrange multiplier fictitious domain dlm fd formulation of glowinski et al int j multiphase flow 25 1999 755 for the fluid rigid body interactions to deal with the fluid flexible body interactions by replacing newton s equations of motion for the rigid body with the continuum equations for the general solid material similar to the rigid body case where the dlm is introduced as a pseudo body force to enforce the constraint of rigid body motion of the fictitious fluid in the solid domain the lagrange multiplier in our formulation is to enforce the fictitious fluid to move at the same velocity as the solid for our computational scheme a first order accurate fractional step scheme is employed to decouple the entire system into three sub systems a fluid problem a solid problem and a lagrange multiplier problem the flow problem is solved with the projection method on half staggered grids the solid problem is solved with the lagrangian finite element method and the newton iterative method and the incompressibility of the material is implemented with the penalty function method the proposed method is applied to two typical fluid structure interaction problems the flow driven oscillation of a flexible plate along the flow direction and the self sustained oscillation across the flow direction both results compare favorably with previously reported numerical and experimental results and show that our method is suited to the simulation of the motion of an incompressible non linear elastic material in a fluid

mixed isogeometric finite cell methods for the stokes problem we study the application of the isogeometric finite cell method iga fcm to mixed formulations in the context of the stokes problem we investigate the performance of the iga fcm when utilizing some isogeometric mixed finite elements namely taylor hood sub grid raviart thomas and nedelec elements these element families have been demonstrated to perform well in the case of conforming meshes but their applicability in the cut cell context is still unclear dirichlet boundary conditions are imposed by nitsche s method numerical test problems are performed with a detailed study of the discrete inf sup stability constants and of the convergence behavior under uniform mesh refinement

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

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

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

harnack inequality for degenerate and singular operators of p laplacian type on riemannian manifolds we study viscosity solutions to degenerate and singular elliptic equations of p laplacian type on riemannian manifolds the krylov safonov type harnack inequality for the p laplacian operators with 1 p infty is established on the manifolds with ricci curvature bounded from below based on abp type estimates we also prove the harnack inequality for nonlinear p laplacian type operators assuming that a nonlinear perturbation of ricci curvature is bounded below

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

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

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

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

an unfitted formulation for the interaction of an incompressible fluid with a thick structure via an xfem dg approach a numerical procedure that combines an extended finite element formulation and a discontinuous galerkin technique is presented with the final aim of providing an effective tool for the simulation of three dimensional 3d fluid structure interaction problems in this work we consider a thick structure immersed in a fluid we describe the numerical models and discuss the specific implementation issues arising in three dimensions finally 3d numerical results are provided to show the effectiveness of the approach

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

convergence of an adaptive discontinuous galerkin method for elliptic interface problems abstract we prove a basic error contraction result of an adaptive discontinuous galerkin method for an elliptic interface problem the interface conditions considered model mass transfer of solutes through semi permeable membranes and other filtering processes the adaptive algorithm is based on a residual type a posteriori error estimator with a bulk refinement criterion the a posteriori error bound is derived under the assumption that the triangulation is aligned with the interfaces although crucially extremely general curved element shapes are also allowed resolving the interface geometry exactly as a corollary convergence of the adaptive discontinuous galerkin method for non essential neumann and or robin type boundary conditions posed on general curved boundaries also follows numerical experiments are also presented

the nitsche method applied to a class of mixed dimensional coupling problems abstract a computational approach for the mixed dimensional modeling of time harmonic waves in elastic structures is proposed a two dimensional 2d structure is considered that includes a part which is assumed to behave in a one dimensional 1d way the 2d and 1d structural regions are discretized using 2d and 1d finite element formulations the coupling of the 2d and 1d regions is performed weakly by using the nitsche method the advantage of using the nitsche method to impose boundary and interface conditions has been demonstrated by various authors here this advantage is shown in the context of mixed dimensional coupling the computational aspects of the method are discussed and it is compared to the slightly simpler penalty method both theoretically and numerically numerical examples are presented in various configurations where the 1d model is either confined laterally or laterally free and where the 2d part is either simply connected or doubly connected the performance is investigated for various wave numbers and various extents of the 1d region varying material properties and distributed loads in the 1d and 2d parts are also considered it is concluded that the nitsche method is a viable technique for mixed dimensional coupling of elliptic problems of this type

analysis and computation of a least squares method for consistent mesh tying in the finite element method a standard approach to mesh tying is to apply lagrange multipliers if the interface is curved however discretization generally leads to adjoining surfaces that do not coincide spatially straightforward lagrange multiplier methods lead to discrete formulations failing a first order patch test t a laursen m w heinstein consistent mesh tying methods for topologically distinct discretized surfaces in non linear solid mechanics internat j numer methods eng 57 2003 1197 1242 this paper presents a theoretical and computational study of a least squares method for mesh tying p bochev d m day a least squares method for consistent mesh tying internat j numer anal modeling 4 2007 342 352 applied to the partial differential equation email protected email protected email protected f f we prove optimal convergence rates for domains represented as overlapping subdomains and show that the least squares method passes a patch test of the order of the finite element space by construction to apply the method to subdomain configurations with gaps and overlaps we use interface perturbations to eliminate the gaps theoretical error estimates are illustrated by numerical experiments

integrating strong and weak discontinuities without integration subcells and example applications in an xfem gfem framework partition of unity methods such as the extended finite element method allows discontinuities to be simulated independently of the mesh int j numer meth engng 1999 45 601 620 this eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re meshing as the discontinuity evolves however to compute the stiffness matrix of the elements intersected by the discontinuity a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted in this paper we use a simple integration technique proposed for polygonal domains int j nuttier meth engng 2009 80 1 103 134 doi 10 1002 nme 2589 to suppress the need for element subdivision numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi material problem show that the proposed method yields accurate results owing to its simplicity the proposed integration technique can be easily integrated in any existing code copyright c 2010 john wiley sons ltd

a stabilized nitsche cut element method for the wave equation abstract we give a weak formulation for solving the wave equation u 2 u f on a 2 dimensional immersed domain in the spatial finite element discretization boundaries do not conform to element boundaries dirichlet and neumann boundary conditions are enforced weakly by nitsche s method additional penalty terms act on the gradient jumps over the interior faces of the elements cut by the boundary these terms ensure a non stiff temporal system which makes it possible to perform explicit time stepping we give optimal a priori error estimates second order accuracy for u u h and u u h and first order accuracy for u u h in l 2 norm numerical results verify this

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

volumetric coupling approaches for multiphysics simulations on non matching meshes summary in finite element analysis of volume coupled multiphysics different meshes for the involved physical fields are often highly desirable in terms of solution accuracy and computational costs we present a general methodology for volumetric coupling of different meshes within a monolithic solution scheme a straightforward collocation approach is compared to a mortar based method for nodal information transfer for the latter dual shape functions based on the biorthogonality concept are used to build the projection matrices thus further reducing the evaluation costs we give a detailed explanation of the integration scheme and the construction of dual shape functions for general first order and second order langrangian finite elements within the mortar method as well as an analysis of the conservation properties of the projection operators moreover possible incompatibilities due to different geometric approximations of curved boundaries are discussed numerical examples demonstrate the flexibility of the presented mortar approach for arbitrary finite element combinations in two and three dimensions and its applicability to different multiphysics coupling scenarios copyright 2016 john wiley sons ltd

the fixed mesh ale approach applied to solid mechanics and fluid structure interaction problems in this paper we propose a method to solve solid mechanics and fluid structure interaction problems using always a fixed background mesh for the spatial discretization the main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves to achieve this we use an arbitrary lagrangian eulerian ale framework the distinctive characteristic being that at each time step results are projected onto a fixed background mesh for solid mechanics problems subject to large strains the fixed mesh fm ale method avoids the element stretching found in fully lagrangian approaches for fsi problems fm ale allows for the use of a single background mesh to solve both the fluid and the structure copyright 2009 john wiley sons ltd

interface handling for three dimensional higher order xfem computations in fluid structure interaction three dimensional higher order extended finite element method xfem computations still pose challenging computational geometry problems especially for moving interfaces this paper provides a method for the localization of a higher order interface finite element fe mesh in an underlying three dimensional higher order fe mesh additionally it demonstrates how a subtetrahedralization of an intersected element can be obtained which preserves the possibly curved interface and allows therefore exact numerical integration the proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their extended axis aligned bounding boxes the intersection method is applied to a highly reduced number of intersection candidates the resulting linearized interface is used as input for an elementwise constrained delaunay tetrahedralization which computes an appropriate subdivision for each intersected element the curved interface is recovered from the linearized interface in the last step the output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element application of the interface algorithm currently concentrates on fluid structure interaction problems on low order and higher order fe meshes which may be composed of any arbitrary element types such as hexahedra tetrahedra wedges etc nevertheless other xfem problems with explicitly given interfaces or discontinuities may be tackled in addition multiple structures and interfaces per intersected element can be handled without any additional difficulties several parallelization strategies exist depending on the desired domain decomposition approach numerical test cases including various geometrical exceptions demonstrate the accuracy robustness and efficiency of the interface handling

an accurate robust and easy to implement method for integration over arbitrary polyhedra application to embedded interface methods we present an accurate method for the numerical integration of polynomials over arbitrary polyhedra using the divergence theorem the method transforms the domain integral into integrals evaluated over the facets of the polyhedra the necessity of performing symbolic computation during such transformation is eliminated by using one dimensional gauss quadrature rule the facet integrals are computed with the help of quadratures available for triangles and quadrilaterals numerical examples in which the proposed method is used to integrate the weak form of the navier stokes equations in an embedded interface method eim are presented the results show that our method is as accurate and generalized as the most widely used volume decomposition based methods moreover since the method involves neither volume decomposition nor symbolic computations it is much easier for computer implementation also the present method is more efficient than other available integration methods based on the divergence theorem efficiency of the method is also compared with the volume decomposition based methods and moment fitting methods to our knowledge this is the first article that compares both accuracy and computational efficiency of methods relying on volume decomposition and those based on the divergence theorem

the arlequin method as a flexible engineering design tool by superposing and gluing models the arlequin method offers an extended modelling framework for the design of engineering structures this paper aims at developing the numerical aspects of the approach and at showing how it can be used with great flexibility and in a consistent manner to change locally a global mechanical model the capabilities of the arlequin method and the effectiveness of the implemented numerical tools are exemplified by 1 d 2 d and 3 d numerical applications

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