least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

global solutions of the super critical 2d quasi geostrophic equation in besov spaces abstract in this paper we study the super critical 2d dissipative quasi geostrophic equation we obtain some regularization effects allowing us to prove a global well posedness result for small initial data lying in critical besov spaces constructed over lebesgue spaces l p with p 1 local results for arbitrary initial data are also given

a residual based a posteriori error estimator for the stokes darcy coupled problem in this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow the flows are governed by the stokes and darcy equations respectively and the transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law the finite element subspaces consider bernardi raugel and raviart thomas elements for the velocities piecewise constants for the pressures and continuous piecewise linear elements for a lagrange multiplier defined on the interface we derive a reliable and efficient residual based a posteriori error estimator for this coupled problem the proof of reliability makes use of suitable auxiliary problems diverse continuous inf sup conditions satisfied by the bilinear forms involved and local approximation properties of the clement interpolant and raviart thomas operator on the other hand helmholtz decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are the main tools for proving the efficiency of the estimator up to minor modifications our analysis can be extended to other finite element subspaces yielding a stable galerkin scheme

coupling fluid flow with porous media flow the transport of substances back and forth between surface water and groundwater is a very serious problem we study herein the mathematical model of this setting consisting of the stokes equations in the fluid region coupled with the darcy equations in the porous medium coupled across the interface by the beavers joseph saffman conditions we prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems this is important because there are many legacy codes available which have been optimized for uncoupled porous media and fluid flow

asymptotic analysis of the laminar viscous flow over a porous bed we consider the laminar viscous channel flow over a porous surface the size of the pores is much smaller than the size of the channel and it is important to determine the effective boundary conditions at the porous surface we study the corresponding boundary layers and by a rigorous asymptotic expansion we obtain saffman s modification of the interface condition observed by beavers and joseph the effective coefficient in the law is determined through an auxiliary boundary layer type problem whose computational and modeling aspects are discussed in detail furthermore the approximation errors for the velocity and for the effective mass flow are given as powers of the characteristic pore size ep finally we give the interface condition linking the effective pressure fields in the porous medium and in the channel and we determine the jump of the effective pressures explicitly

error analysis for constrained first order system least squares finite element methods in this paper a general error analysis is provided for finite element discretizations of partial differential equations in a saddle point form with divergence constraint in particular this extends upon the work of j h adler and p s vassilevski springer proc math statist 45 springer new york 2013 pp 1 19 giving a general error estimate for finite element problems augmented with a divergence constraint and showing that these estimates are obtained for problems such as diffusion and stokes using the first order system least squares fosls finite element method the main result is that by enforcing the constraint on a mathbf h 1 equivalent fosls formulation one maintains optimal convergence of the fosls functional i e the energy norm of the error while guaranteeing the conservation of the divergence constraint i e mass conservation in some examples the error estimates and results depend on using finite elements for the constraint space that are inf sup stable when paired wi

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

global well posedness for a modified critical dissipative quasi geostrophic equation in this paper we consider the following modified quasi geostrophic equation t u d 0 u d 1r x r2 with 0 and 0 1 1 2 when 0 1 the equation was firstly introduced by constantin iyer and wu 2008 in 11 here by using the modulus of continuity method we prove the global well posedness of the system as a byproduct we also show that for every 0 2 the lipschitz norm of the solution has a uniform exponential upper bound

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

a crank nicolson discontinuous finite volume element method for a coupled non stationary stokes darcy problem abstract in this paper we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a crank nicolson discretization in time and we obtain the solution at the first time step by applying a first order backward euler method to solve the coupling of time dependent stokes darcy problem the proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head whereas the pressure is approximated by the piecewise constant elements then the unique solvability of the approximate solution for the discrete problem is derived in addition the optimal error estimates of the full discretization in standard l 2 norm and broken h 1 norm are obtained finally a series of numerical experiments are provided to illustrate the features of the proposed method such as the optimal accuracy orders adaptive mesh refinement mass conservation capability to conveniently deal with complicated geometries with different types of boundary conditions and the applicability to the problems with realistic parameters in particular the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain it encompasses both fractures with high permeability which act as the prior flow conduit and fractures with low permeability as a flow barrier

global well posedness and blow up of solutions for the camassa holm equations with fractional dissipation consideration in this paper is the effect of varying fractional dissipation with the dissipative operator power gamma ge 0 on the well posedness of the camassa holm equations with fractional dissipation it is shown that the zero filter limit alpha rightarrow 0 of the camassa holm equation with fractional dissipation is the fractal burgers equation it is known that in the supercritical case gamma in 0 1 the fractal burgers equation blows up in finite time in h s mathbb r with s frac 3 2 gamma it is established here that the dissipative camassa holm equation is globally well posed in the critical sobolev space h frac 3 2 gamma mathbb r with the fractional parameter gamma in frac 1 2 1 moreover it is also demonstrated that the solution of the dissipative camassa holm equation blows up in finite time in the particular supercritical case when gamma 0

least squares for second order elliptic problems abstract in this paper we introduce and analyze two least squares methods for second order elliptic differential equations with mixed boundary conditions these methods extend to problems which involve oblique derivative boundary conditions as well as nonsymmetric and indefinite problems as long as the original problem has a unique solution with the methods to be developed neumann and oblique boundary conditions are imposed weakly and thus avoid compatibility conditions on the finite element subspaces the resulting least squares approximations are unconditionally stable no conditions on the step size h and will be shown to converge at an optimal rate the first least squares method involves a discrete computable h 1 norm of the residual and stabilization terms consisting of the jumps at the interelement boundaries and a weighted elementwise l 2 norm of the residual over the finite elements this method is developed without the introduction of additional problem variables the second method involves the use of the flux as an additional unknown although this method is similar to the least squares method for first order systems introduced in 7 it differs in that discontinuous finite elements are allowed it is also more general in that it extends to the oblique boundary problem

a weak galerkin finite element method for a coupled stokes darcy problem on general meshes abstract in this paper we introduce and analyze a weak galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow flows are governed by the stokes equations in primal velocity pressure formulation and darcy equation in the second order primary formulation respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law by using the weak galerkin approach we consider the two dimensional problem with the usual polynomials of degree k 1 for the velocity and hydraulic head while polynomials of degree k 1 for the pressure the velocity and hydraulic head is enhanced by polynomials of degree k 1 on the edge of a finite element partition this new method has a lot of attractive computational features more general finite element partitions of arbitrary polygons with certain shape regularity fewer degrees of freedom and parameter free stability and error estimates of optimal order are obtained moreover numerical experiences are presented to illustrate the good performance confirm the optimal order of convergence and verify the efficiency of the proposed weak galerkin method in this paper we also deal with finite element partitions consisting of general meshes such as triangular mesh quadrilateral mesh hexagonal dominant mesh and voronoi mesh for the numerical weak galerkin approximation

a domain decomposition method for the steady state navier stokes darcy model with beavers joseph interface condition this paper proposes and analyzes a robin type multiphysics domain decomposition method ddm for the steady state navier stokes darcy model with three interface conditions in addition to the two regular interface conditions for the mass conservation and the force balance the beavers joseph condition is used as the interface condition in the tangential direction the major mathematical difficulty in adopting the beavers joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system y cao et al comm math sci 8 2010 pp 1 25 y cao et al siam j numer anal 47 2010 pp 4239 4256 in this paper the well posedness of the navier stokes darcy model with beavers joseph condition is analyzed by using a branch of nonsingular solutions by following the idea in y cao et al numer math 117 2011 pp 601 629 the three physical interface conditions are utilized together to construct the robin type boundary conditions on th

a new bernstein s inequality and the 2d dissipative quasi geostrophic equation we show a new bernstein s inequality which generalizes the results of cannone planchon danchin and lemarie rieusset as an application of this inequality we prove the global well posedness of the 2d quasi geostrophic equation with the critical and super critical dissipation for the small initial data in the critical besov space and local well posedness for the large initial data

shape calculus and finite element method in smooth domains the use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary hence the approximation error consists of two parts the geometric part and the finite element part we propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain the finite element part of the error can be treated in the standard way to estimate the geometric part of the error we need quantitative estimates related to perturbation of the geometry we derive such estimates using the techniques developed for shape sensitivity analysis

accuracy enhancement for non isoparametric finite element simulations in curved domains application to fluid flow abstract among a few known techniques the isoparametric version of the finite element method for meshes consisting of curved triangles or tetrahedra is the one most widely employed to solve pdes with essential conditions prescribed on curved boundaries it allows to recover optimal approximation properties that hold for elements of order greater than one in the energy norm for polytopic domains however besides a geometric complexity this technique requires the manipulation of rational functions and the use of numerical integration we consider a simple alternative to deal with dirichlet boundary conditions that bypasses these drawbacks without eroding qualitative approximation properties in the present work we first recall the main principle this technique is based upon by taking as a model the solution of the poisson equation with quadratic lagrange finite elements then we show that it extends very naturally to viscous incompressible flow problems although the technique applies to any higher order velocity pressure pairing as an illustration a thorough study thereof is conducted in the framework of the stokes system solved by the classical taylor hood method

asymptotic expansion of solutions to the drift diffusion equation with fractional dissipation abstract the initial value problem for the drift diffusion equation arising from the model of semiconductor device simulations is studied the dissipation on this equation is given by the fractional laplacian 2 large time behavior of solutions to the drift diffusion equation with 0 1 is discussed when 1 large time behavior of solutions is known however when 0 1 the perturbation methods used in the preceding works would not work in this paper the asymptotic expansion of solutions with high order is derived

remarks on the method of modulus of continuity and the modified dissipative porous media equation abstract we employ besov space techniques and the method of modulus of continuity to obtain the global well posedness of the modified porous media equation

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

optimal finite element interpolation on curved domains this paper is devoted to a general theory of approximation of functions in finite element spaces in particular the case is considered where the functions to be interpolated are on the one hand not very smooth and on the other are defined on curved domains thus a number of well known optimal interpolation results are generalized

well posedness of the cauchy problem for the fractional power dissipative equations abstract this paper studies the cauchy problem for the nonlinear fractional power dissipative equation u t u f u for initial data in the lebesgue space l r r n with either r r d n b 2 d or the homogeneous besov space b p r n with 2 d b n p and 1 p where 0 f u f u or q d f u with q d being a homogeneous pseudo differential operator of order d 0 2 and f u is a function of u which behaves like u b u with b 0

nonconforming elements in least squares mixed finite element methods in this paper we analyze the finite element discretization for the first order system least squares mixed model for the second order elliptic problem by means of using nonconforming and conforming elements to approximate displacement and stress respectively moreover on arbitrary regular quadrilaterals we propose new variants of both the rotated q1 nonconforming element and the lowest order raviart thomas element

first order system least squares for the signorini contact problem in linear elasticity a first order system least squares formulation for the signorini problem modeling frictionless contact in linear elasticity is studied in addition to the displacement field the stress tensor is used as an independent process variable a contact boundary term is added to the usual least squares functional in order to achieve coercivity and continuity in appropriate norms the discrete functional is shown to constitute an a posteriori error estimator on which an adaptive refinement strategy may be based as finite element spaces standard conforming piecewise polynomials for the displacement approximation are combined with raviart thomas elements for the rows in the stress tensor computational results for a test problem of hertzian contact illustrate the effectiveness of our least squares approach

least squares methods for elliptic systems a weighted least squares method is given for the numerical solution of elliptic partial differential equations of agmon douglis nirenberg type and an error analysis is provided some examples are given 1 introduction the use of least squares methods for the approximate solution of equations dates back at least to gauss the modern theory of least squares methods in the numerical solution of elliptic boundary value problems starts in 1970 with the papers of bramble and schatz 5 6 this work uses a finite dimensional space s of approximating functions similar to the spaces used in finite element methods the approximate solution is defined to be the minimizer of a least squares functional that is a weighted sum of the least squares residual in the differential equation and the least squares residual in the boundary condition the paper 5 has an historical importance for the following reason it appeared during the time when numerical analysts were shifting attention from finite difference methods to finite element methods and it provided for the first time a family of approximation methods for the solution of the dirichlet problem whose order of accuracy could be made arbitrarily large the paper 6 provided an extension to an elliptic equation of order 2m and 3 gave important simplifications in the analysis the principal advantages of the method are that one need not satisfy exactly the dirichlet boundary condi tions and that the mathematical analysis dictates in a natural way the relative weights that are given to the boundary and interior terms in the least squares functional also the method provides in a quasioptimal sense as good a solution as can be expected from the space s on the other hand the method requires that s consist of functions which are smooth enough to lie in the domain of the elliptic operator also the method seems to produce matrices with large condition number for various reasons it is of interest to extend the theory of least squares methods to include elliptic systems first if a second order elliptic equation is written as a first order system it would seem and this is borne out by our analysis that the smoothness requirements for the spaces of approximating functions would be reduced thus eliminating one of the disadvantages of the method a second motivation for extending the least squares method to elliptic systems is that elliptic systems occur frequently in applications an example of an elliptic system is the system of equations for stokes flow for this system the least squares method does not require the space of approximating vector fields to be incompressible instead

finite element analysis of varitional crimes for a quasilinear elliptic problem in 3d we examine a finite element approximation of a quasilinear boundary value elliptic problem in a three dimensional bounded convex domain with a smooth boundary the domain is approximated by a polyhedron and a numerical integration is taken into account we apply linear tetrahedral finite elements and prove the convergence of approximate solutions on polyhedral domains in the w 1 2 norm to the true solution without any additional regularity assumptions

l 2 projected least squares finite element methods for the stokes equations two new l2 least squares ls finite element methods are developed for the velocity pressure vorticity first order system of the stokes problem with dirichlet velocity boundary condition a key feature of these new methods is that a local or almost local l2 projector is applied to the residual of the momentum equation such l2 projection is always defined onto the linear finite element space no matter which finite element spaces are used for velocity pressure vorticity variables consequently the implementation of this l2 projected ls method is almost as easy as that of the standard l2 ls method more importantly the former has optimal error estimates in l2 norm with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal order interpolations and for all three variables velocity pressure and vorticity using unequal order interpolations numerical experiments are given to demonstrate the theoretical results

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

global well posedness of incompressible flow in porous media with critical diffusion in besov spaces abstract in this paper we study the model of heat transfer in a porous medium with a critical diffusion we obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in besov spaces b p 1 3 p r 3 with 1 p by the method of modulus of continuity and fourier localization technique

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

on discretely entropy stable weight adjusted discontinuous galerkin methods curvilinear meshes abstract we construct entropy conservative and entropy stable discontinuous galerkin dg discretizations for time dependent nonlinear hyperbolic conservation laws on curvilinear meshes the resulting schemes preserve a semi discrete quadrature approximation of a continuous global entropy inequality the proof requires the satisfaction of a discrete geometric conservation law which we enforce through an appropriate polynomial approximation we extend the construction of entropy conservative and entropy stable dg schemes to the case when high order curvilinear mass matrices are approximated using low storage weight adjusted approximations and describe how to retain global conservation properties under such an approximation for certain types of curvilinear meshes these weight adjusted approximations deliver optimal rates of convergence finally the high order accuracy local conservation and discrete conservation or dissipation of entropy for under resolved solutions are verified through numerical experiments for the compressible euler equations on triangular and tetrahedral meshes

natural superconvergence points in three dimensional finite elements a systematic and analytic process is conducted to identify natural superconvergence points of high degree polynomial c 0 finite elements in a three dimensional setting this identification is based upon explicitly constructing an orthogonal decomposition of local finite element spaces derivative and function value superconvergence points are investigated for both the poisson and the laplace equations superconvergence results are reported for hexahedral pentahedral and tetrahedral elements up to certain degrees

first order system least squares for coupled stokes darcy flow the coupled problem with stokes flow in one subdomain and a darcy flow model in a second subdomain is studied in this paper both flow problems are treated as first order systems involving pseudostress and velocity in the stokes case and using a flux pressure formulation in the darcy subdomain as process variables respectively the beavers joseph saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems a combination of h mathrm div conforming raviart thomas and standard h 1 conforming elements is used for the stokes as well as for the darcy subsystem the homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates it also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies

analysis of the fully discrete fat boundary method the fat boundary method is a method of the fictitious domain class which was proposed to solve elliptic problems in complex geometries with non conforming meshes it has been designed to recover optimal convergence at any order despite of the non conformity of the mesh and without any change in the discrete laplace operator on the simple shape domain we propose here a detailed proof of this high order convergence and propose some numerical tests to illustrate the actual behaviour of the method

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

global solution of the critical burgers equation in n dimensions abstract in this paper we first prove the local well posedness of the fractional burgers equations in n dimensions combining the local well posedness and the method of modulus of continuity we show the global well posedness of the n d critical burgers equation in critical besov spaces b p 1 n p r n with p n

parametric raviart thomas elements for mixed methods on domains with curved surfaces the finite element approximation on curved boundaries using parametric raviart thomas spaces is studied in the context of the mixed formulation of poisson s equation as a saddle point system it is shown that optimal order convergence is retained on domains with piecewise c k 2 boundary for the parametric raviart thomas space of degree k geq 0 under the usual regularity assumptions this extends the analysis in f bertrand s munzenmaier and g starke siam j numer anal 52 2014 pp 3165 3180 from the first order system least squares formulation to mixed approaches of saddle point type in addition a detailed proof of the crucial estimate in three dimensions is given which handles some complications not present in the two dimensional case moreover the appropriate treatment of inhomogeneous flux boundary conditions is discussed the results are confirmed by computational results which also demonstrate that optimal order convergence is not achieved in general if standard raviart th

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

least squares mixed finite element solution of variably saturated subsurface flow problems a least squares mixed finite element formulation is applied to the nonlinear elliptic problems arising in each time step of an implicit euler discretization for variably saturated flow problems this approach simultaneously constructs approximations to the flux in raviart thomas spaces and to the hydraulic potential by standard h1 conforming linear finite elements two important properties of the least squares approach are investigated in detail the local least squares functional provides an a posteriori error estimator and gauss newton methods are robust iterative solvers for the resulting nonlinear least squares problems computational experiments conducted for a realistic water table recharge problem illustrate the effectiveness of this approach

first order system least squares on curved boundaries higher order raviart thomas elements with this paper our investigation of the finite element approximation on curved boundaries using raviart thomas spaces in the context of first order system least squares methods is continued and extended to the higher order case it is shown that the optimal order of convergence is retained from the lowest order case if a parametric version of raviart thomas elements is used this is illustrated numerically for an elliptic boundary value problem involving a circular boundary curve

first order system least squares method for the interface problem of the stokes equations the first order system least squares method for the stokes equation with discontinuous viscosity and singular force along the interface is proposed and analyzed first interface conditions are derived by introducing a physical meaningful variable such as the velocity gradient the stokes equation transformed into a first order system of equations then the continuous and discrete norm least squares functionals using legendre and chebyshev weights for the first order system are defined we showed that continuous and discrete homogeneous least squares functionals are equivalent to appropriate product norms the spectral convergence of the proposed method is given a numerical example is provided to support the method and its analysis

an estimate for the three dimensional discrete green s function and applications in this paper we first introduce definitions of the regularized green s function and the discrete green s function in three dimensions with complicated arguments estimates for these two functions are then derived finally we give an application of the estimate for the discrete green s function to the high accuracy analysis of the three dimensional block finite element approximation

nurbs based least squares finite element methods for fluid and solid mechanics summary this contribution investigates the performance of a least squares finite element method based on non uniform rational b splines nurbs basis functions the least squares functional is formulated directly in terms of the strong form of the governing equations and boundary conditions thus the introduction of auxiliary variables is avoided but the order of the basis functions must be higher or equal to the order of the highest spatial derivatives the methodology is applied to the incompressible navier stokes equations and to linear as well as nonlinear elastic solid mechanics the numerical examples presented feature convective effects and incompressible or nearly incompressible material the numerical results which are obtained with equal order interpolation and without any stabilisation techniques are smooth and accurate it is shown that for p and h refinement the theoretical rates of convergence are achieved copyright 2014 john wiley sons ltd

adaptive least squares finite element methods for linear transport equations based on an h div flux reformulation abstract in this paper we study the least squares finite element methods lsfem for the linear hyperbolic transport equations the linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions while the normal component of the flux across the mesh faces needs to be continuous traditional lsfems using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions in order to separate the continuity requirements a new flux variable is introduced with this reformulation the continuities of the flux and the solution can be handled separately in natural h div l 2 conforming finite element spaces several variants of the methods are developed to handle the inflow boundary condition strongly or weakly with the reformulation the new lsfems can handle discontinuous solutions and boundary conditions much better than the traditional lsfems with continuous polynomial approximations with least squares functionals as a posteriori error estimators the adaptive methods can naturally identify error sources including singularity and non matching discontinuity for discontinuity aligned mesh no extra error is introduced if an r t 0 p 0 pair is used to approximate the flux and the solution the new adaptive lsfems can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods extensive numerical tests are performed to show the effectiveness of the methods developed in the paper

global well posedness of the critical burgers equation in critical besov spaces abstract we make use of the method of modulus of continuity a kiselev f nazarov r shterenberg blow up and regularity for fractal burgers equation dyn partial differ equ 5 2008 211 240 and fourier localization technique h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 to prove the global well posedness of the critical burgers equation t u u x u u 0 in critical besov spaces b p 1 1 p r with p 1 where

penalty method with p1 p1 finite element approximation for the stokes equations under the slip boundary condition we consider the p1 p1 or p1b p1 finite element approximations to the stokes equations in a bounded smooth domain subject to the slip boundary condition a penalty method is applied to address the essential boundary condition u cdot n g on partial omega which avoids a variational crime and simultaneously facilitates the numerical implementation we give o h 1 2 epsilon 1 2 h epsilon 1 2 error estimate for velocity and pressure in the energy norm where h and epsilon denote the discretization parameter and the penalty parameter respectively in the two dimensional case it is improved to o h epsilon 1 2 h 2 epsilon 1 2 by applying reduced order numerical integration to the penalty term the theoretical results are confirmed by numerical experiments

solving elliptic finite element systems in near linear time with support preconditioners we consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems our main result is that these linear systems which are symmetric and positive semidefinite are well approximated by symmetric diagonally dominant matrices our framework for defining matrix approximation is support theory significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case and in particular it is known that such systems can be solved with iterative methods in nearly linear time thus our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time we show that the support number bounds which control the number of iterations in the preconditioned iterative solver depend on mesh quality measures but not on the problem size or shape of the domain

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

well posedness of the cauchy problem for the fractional power dissipative equation in critical besov spaces abstract in this paper we study the cauchy problem for the semilinear fractional power dissipative equation u t u f u for the initial data u 0 in critical besov spaces b 2 r with n 2 2 d b where 0 f u p d u b 1 with p d being a homogeneous pseudo differential operator of order d 0 2 and b 0 being an integer making use of some estimates of the corresponding linear equation in the frame of mixed time space spaces the so called mono norm method which is different from the kato s double norm method fourier localization technique and littlewood paley theory we get the well posedness result in the case n 2

comparison of high order curved finite elements several finite element techniques used in domains with curved boundaries are discussed and compared with particular emphasis in two issues the exact boundary representation of the domain and the consistency of the approximation the influence of the number of integration points on the accuracy of the computation is also studied two dimensional numerical examples solved with continuous and discontinuous galerkin formulations are used to test and compare all these methodologies in every example shown the recently proposed nurbs enhanced finite element method nefem provides the maximum accuracy for a given spatial discretization at least one order of magnitude more accurate than classical isoparametric finite element method fem moreover nefem outperforms cartesian fem and p fem stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations

analysis of the domain mapping method for elliptic diffusion problems on random domains in this article we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains in particular based on the decay of the karhunen loeve expansion of the domain perturbation field we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension for the implementation of a related approximation scheme like quasi monte carlo quadrature stochastic collocation etc we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field this simplifies the implementation and yields a non intrusive approach having this machinery at hand we can easily transfer it to stochastic interface problems the theoretical findings are complemented by numerical examples for both stochastic interface problems and boundary value problems on random domains

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

shape optimization of peristaltic pumping transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this it is also important to many industrial processes we present a variational method for optimizing the wave shape of a peristaltic pump specifically we optimize the wave profile of a two dimensional channel containing a navier stokes fluid with no assumption on the wave profile other than it is a traveling wave e g we do not assume it is the graph of a function hence this is an infinite dimensional optimization problem the optimization criteria consists of minimizing the input fluid power due to the peristaltic wave subject to constraints on the average flux of fluid and area of the channel sensitivities of the cost and constraints are computed variationally via shape differential calculus and we use a sequential quadratic programming sqp method to find a solution of the first order kkt conditions we also use a merit function based line search in order to balance between decreasing the cost and keeping the constraints satisfied when updating the channel shape our numerical implementation uses a finite element method for computing a solution of the navier stokes equations adjoint equations as well as for the sqp method when computing perturbations of the channel shape the walls of the channel are deformed by an explicit front tracking approach in computing functional sensitivities with respect to shape we use l 2 type projections for computing boundary stresses and for geometric quantities such as the tangent field on the channel walls and the curvature we show error estimates for the boundary stress and tangent field approximations as a result we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature specifically we see highly asymmetric wave shapes that are far from being sine waves many examples are shown for a range of fluxes and reynolds numbers up to re 500 which illustrate the capabilities of our method

the optimal refinement strategy for 3 d simplicial meshes 3 d mesh refinement strategies are the object of interest in the present paper the dependence of finite element solutions on various refinement procedures is discussed a new refinement strategy which is optimal with respect to the measure of degeneracy for all canonical domains is obtained comparisons with known refinement techniques are made some isoparametric examples are presented

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

multilevel boundary functionals for least squares mixed finite element methods for least squares mixed finite element methods for the first order system formulation of second order elliptic problems a technique for the weak enforcement of boundary conditions is presented this approach is based on least squares boundary functionals which are equivalent to the h 1 2 and h1 2 norms on the trace spaces of lowest order raviart thomas elements for the flux and standard continuous piecewise linear elements for the pressure respectively continuity and coercivity of the resulting bilinear form is proved implying optimal order convergence of the resulting galerkin approximation the boundary least squares functional is implemented using multilevel principles and the technique is tested numerically for a model problem

finite element approximations for the stokes equations on curved domains and their errors we construct finite element spaces for the stokes equations on curved domains which satisfy the email protected ka condition moreover we estimate the errors of the finite element approximations for the stokes equations

least squares spectral method for the two dimensional stokes interface problems abstract this paper deals with the pseudo spectral least squares method to approximate the solution of the stokes interface problem that is the stokes equations with discontinuous viscosity and singular source term the stokes interface equations are adjusted in each sub domain along with the interface conditions then the equations are reformulated as a first order system by introducing stress tensor as a new independent variable the continuous and discrete norm least squares functionals using legendre and chebyshev weights for the first order system are defined and shown to be equivalent to appropriate product norms the spectral convergence of the proposed method is presented numerical examples are also given to demonstrate the method and analysis

two preconditioners based on the multi level splitting of finite element spaces the hierarchical basis preconditioner and the recent preconditioner of bramble pasciak and xu are derived and analyzed within a joint framework this discussion elucidates the close relationship between both methods special care is devoted to highly nonuniform meshes exclusively local properties like the shape regularity of the finite elements are utilized

decay of solutions to anisotropic conservation laws with large initial data abstract in this paper we study the large time behavior of solutions to the cauchy problem for the anisotropic conservation laws in two dimensional space without any smallness assumption on the initial data the decay rates of solutions in l 2 space and homogeneous sobolev space h are obtained by using the method of time frequency decomposition and the classical energy method

wavelet least squares methods for boundary value problems this paper is concerned with least squares methods for the numerical solution of operator equations our primary focus is the discussion of the following conceptual issues the selection of appropriate least squares functionals their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations we describe first a general format of variational problems that are well posed in a certain natural topology in order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems their formulation as a first order system transmission problems the system of stokes equations or more general saddle point problems particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions we propose a unified treatment based on wavelet expansions in particular we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context this provides access to difficult norms efficient preconditioners and in the case of first order systems optimal l2 error estimates

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives