Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

A priori error analysis of the mortar finite element method for variational inequalities

The mortar finite element method is a special domain decomposition method, which can handle the situation where meshes on different subdomains need not align across the interface. In this article, we will apply the mortar element method to general variational inequalities of free boundary type, such as free seepage flow, which may show different behaviors in different regions. We prove that if the solution of the original variational inequality belongs to H²(D), then the mortar element solution can achieve the same order error estimate as the conforming P₁ finite element solution. Application of the mortar element method to a free surface seepage problem and an obstacle problem verifies not only its convergence property but also its great computational efficiency. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Fast computations with the harmonic Poincaré–Steklov operators on nested refined meshes

In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincaré–Steklov operators (Dirichlet–Neumann mapping) for interior and exterior problems in the presence of a nested mesh refinement. Our approach is based on the multilevel interface solver applied to the Schur complement reduction onto the nested refined interface associated with a nonmatching decomposition of a polygon by rectangular substructures. This paper extends methods from Khoromskij and Prössdorf (1995), where the finite element approximations of interior problems on quasi‐uniform grids have been considered. For both interior and exterior problems, the matrix–vector multiplication with the compressed Schur complement matrix on the interface is shown to have a complexity of the order O(N _r log³ N _u), where N_r = O((1 + p _r) N _u) is the number of degrees of freedom on the polygonal boundary under consideration, N _u is the boundary dimension of a finest quasi‐uniform level and p _r ⩾ 0 defines the refinement depth. The corresponding memory needs are estimated by O(N _r log^q N _u), where q = 2 or q = 3 in the case of interior and exterior problems, respectively.

     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L^∞(L²) norm. Numerical experiment is carried out for an L‐shaped domain to illustrate our theoretical findings.     

Gradient based enhanced finite element formulation for diffuse phase interfaces

Interfaces between adjacent phases, so-called domain walls, appear as non-linear gradients of order parameters in diffuse phase field models. Usually, the interface width is much smaller than the dimension of the simulated region. Since the position of domain walls is not known a priori the maximum size of finite elements needs to be adapted to the length scale of interfaces within the entire region. We suggested a selective finite element method to improve the numerical solution of diffuse phase field models [1, 2]. It enhances the finite element interpolation space using supplementary local degrees of freedom. However, corresponding additional nodes are strictly located in the interior of elements, thus, C⁰-continuity at element border is guaranteed. Since C⁰-continuity limits the performance of this method we propose in this paper a relaxation of C⁰-requirements perpendicular to the gradient of the order parameter. Therefore, the direction of interfaces is analyzed as additional information for further adaptive improvement of the interpolation space. A dual phase field model is used to validate the proposed method. The analytical solution of a stationary domain wall allows error analysis of regular and distorted finite element meshes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a kinematic dynamo model with FEM–BEM coupling

We consider a kinematic dynamo model in a bounded interior simply connected region Ω and in an insulating exterior region . In the so‐called direct problem, the magnetic field B and the electric field E are unknown and are driven by a given incompressible flow field w . After eliminating E , a vector and a scalar potential ansatz for B in the interior and exterior domains, respectively, are applied, leading to a coupled interface problem. We apply a finite element approach in the bounded interior domain Ω, whereas a symmetric boundary element approach in the unbounded exterior domain Ω^c is used. We present results on the well‐posedness of the continuous coupled variational formulation, prove the well‐posedness and stability of the semi‐discretized and fully discretized schemes, and provide quasi‐optimal error estimates for the fully discretized scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the least squares solution of the groundwater flow equation

The least squares finite element method is a member of the weighted residuals class of numerical methods for solving partial differential equations. The least squares finite element method is applied to the groundwater flow equation. Space is discretized with a C¹ continuous trial function and parameters are approximated with a C⁰ bilinear basis. Solutions for problems containing parameters with large localized spatial gradients are characterized by errors that are propagated throughout the entire domain. Second-order spatial convergence is observed, and extreme mesh refinement is required to match Galerkin and mixed least squares finite element results. Temporal discretization should be kept separate from the least squares spatial discretization. © 1994 John Wiley & Sons, Inc.     

The modified method of characteristics with mixed finite element domain decomposition procedures for the transient behavior of a semiconductor device

For the transient behavior of a semiconductor device, the modified method of characteristics with mixed finite element domain decomposition procedures applicable to parallel arithmetic is put forward. The electric potential equation is described by the mixed finite element method, and the electric, hole concentration and heat conduction equations are treated by the modified method of characteristics finite element domain decomposition methods. Some techniques, such as calculus of variations, domain decomposition, characteristic method, energy method, negative norm estimate and prior estimates and techniques are employed. Optimal order estimates in L² norm are derived for the error in the approximation solution. Thus the well‐known theoretical problem has been thoroughly and completely solved.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 353–368 2012     

Adaptive FE–BE coupling for strongly nonlinear transmission problems with Coulomb friction

We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the p-Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of L ^p - and L ²-Sobolev spaces.     

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L² projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.     

Domain decomposition method and numerical analysis of a fluid dynamics problem

A two-dimensional problem obtained by time discretization and linearization of a viscous flow governed by the incompressible Navier-Stokes equations is considered. The original domain is divided into subdomains such that their interface is a smooth (nonclosed, self-avoiding) curve with the ends belonging to the boundary of the domain. A nonconforming finite element method is constructed for the problem, and the convergence rate of the discrete solution of the problem to the exact one is estimated in the L ₂(Ω _h ) norm.     

Development of a P2 element with optimal L2 convergence for biharmonic equation

It is well known that convergence rate of finite element approximation is suboptimal in the L² norm for solving biharmonic equations when P₂ or Q₂ element is used. The goal of this paper is to derive a weak Galerkin (WG) P₂ element with the L² optimal convergence rate by assuming the exact solution sufficiently smooth. In addition, our new WG finite element method can be applied to general mesh such as hybrid mesh, polygonal mesh or mesh with hanging node. The numerical experiments have been conducted on different meshes including hybrid meshes with mixed of pentagon and rectangle and mixed of hexagon and triangle.     

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

In this paper, the weak Galerkin finite element method (WG-FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG-FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Optimal pointwise-in-time error estimates in L²-norm and H¹-norm are shown to hold for the semidiscrete scheme even if the regularity of the solution is low on the whole domain. Furthermore, a fully discrete approximation based on backward Euler scheme is analyzed and related optimal error estimates are derived.     

A posteriori error estimates for a coupled wave system with a local damping

We study a finite element method applied to a system of coupled wave equations in a bounded smooth domain in \mathbbR^d {\mathbb{R}^d} , d = 1, 2, 3, associated with a locally distributed damping function. We start with a spatially continuous finite element formulation allowing jump discontinuities in time. This approach yields, L ₂(L ₂) and L _∞(L ₂), a posteriori error estimates in terms of weighted residuals of the system. The proof of the a posteriori error estimates is based on the strong stability estimates for the corresponding adjoint equations. Optimal convergence rates are derived upon the maximal available regularity of the exact solution and justified through numerical examples. Bibliography: 14 titles. Illustrations: 4 figures.     

An H‐matrix Type Preconditioner For Frictional Contact Problems

The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non‐overlapping domain decomposition method [2, 3]. The global problem is transformed to a independant local problems posed in each bodie and a problem posed on the contact surface (the interface problem). The solution is obtained by using a successive approximation method, in each step of this algorithm we solve two intermediate problems the first with prescribed tangential pressure and the second with prescribed normal pressure [8]. Our preconditioner construction is based on the application of the H‐matrix technique [6, 7] together with the representation of the H^1/2 seminorm by a sum of partial seminorms [4]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Error estimates for the finite element approximation to a maxwell-type boundary value problem

Maxwell's boundary value problem for the time-harmonic case in a smooth, bounded domain G of R ² is considered. The optimal asymptotic L₂(G) and H₁(G)-error estimates 0(h²) and 0(h) resp, are derived for a piecewise linear finite element solution.     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007