Fast parallel solvers for symmetric boundary element domain decomposition equations

Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of algebraic complexity and of high parallel efficiency, where denotes the usual discretization parameter. Received August 28, 1996 / Revised version received March 10, 1997     

Degenerate two-phase incompressible flow III. Sharp error estimates

Summary. This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system. Received March 9, 1998 / Revised version received July 17, 2000 / Published online May 30, 2001     

Two different approaches for matching nonconforming grids: The Mortar Element method and the Feti Method

When using domain decomposition in a finite element framework for the approximation of second order elliptic or parabolic type problems, it has become appealing to tune the mesh of each subdomain to the local behaviour of the solution. The resulting discretization being then nonconforming, different approaches have been advocated to match the admissible discrete functions. We recall here the basics of two of them, the Mortar Element method and the Finite Element Tearing and Interconnecting (FETI) method, and aim at comparing them. The conclusion, both from the theoretical and numerical point of view, is in favor of the mortar element method.     

Composite finite elements for the approximation of PDEs on domains with complicated micro-structures

Summary. Usually, the minimal dimension of a finite element space is closely related to the geometry of the physical object of interest. This means that sometimes the resolution of small micro-structures in the domain requires an inadequately fine finite element grid from the viewpoint of the desired accuracy. This fact limits also the application of multi-grid methods to practical situations because the condition that the coarsest grid should resolve the physical object often leads to a huge number of unknowns on the coarsest level. We present here a strategy for coarsening finite element spaces independently of the shape of the object. This technique can be used to resolve complicated domains with only few degrees of freedom and to apply multi-grid methods efficiently to PDEs on domains with complex boundary. In this paper we will prove the approximation property of these generalized FE spaces. Received June 9, 1995 / Revised version received February 5, 1996     

Convergence of a penalty-finite element approximation for an obstacle problem

Summary This study establishes an error estimate for a penalty-finite element approximation of the variational inequality obtained by a class of obstacle problems. By special identification of the penalty term, we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality. The rate of convergence of the penalization is where is the penalty parameter. To obtain the error of finite element approximation, we apply the results obtained by Brezzi, Hager and Raviart for the mixed finite element method to the variational inequality.     

A finite element lumped mass scheme for solving eigenvalue problems of circular arches

Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.     

A framework for obtaining guaranteed error bounds for finite element approximations

We give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming, non-conforming, mixed and discontinuous finite element schemes. Whilst the details of the actual estimator are rather different for each particular scheme, there is nonetheless a common underlying structure at work in all cases. We aim to illustrate this structure by treating conforming, non-conforming and discontinuous finite element schemes in a single framework. In taking a rather general viewpoint, some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but, in return, we hope to allow the reader to ‘see the wood despite the trees’.     

Finite element approximationof a nonlinear elliptic equation arising from bimaterial problemsin elastic-plastic mechanics

Summary. In [13], a nonlinear elliptic equation arising from elastic-plastic mechanics is studied. A well-posed weak formulation is established for the equation and some regularity results are further obtained for the solution of the boundary problem. In this work, the finite element approximation of this boundary problem is examined in the framework of [13]. Some error bounds for this approximation are initially established in an energy type quasi-norm, which naturally arises in degenerate problems of this type and proves very useful in deriving sharper error bounds for the finite element approximation of such problems. For sufficiently regular solutions optimal error bounds are then obtained for some fully degenerate cases in energy type norms. Received June 12, 1998 / Revised version received June 21, 1999 / Published online June 8, 2000     

A collocation method for high-frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.     

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Summary. A least-squares mixed finite element method for general second-order non-selfadjoint elliptic problems in two- and three-dimensional domains is formulated and analyzed. The finite element spaces for the primary solution approximation and the flux approximation consist of piecewise polynomials of degree and respectively. The method is mildly nonconforming on the boundary. The cases and are studied. It is proved that the method is not subject to the LBB-condition. Optimal - and -error estimates are derived for regular finite element partitions. Numerical experiments, confirming the theoretical rates of convergence, are presented. Received October 15, 1993 / Revised version received August 2, 1994     

An algorithm using the finite volume element method and its splitting extrapolation

This paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation. This method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation, such as a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than a Richardson extrapolation. Because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition.     

On the asymptotic exactness of error estimators for linear triangular finite elements

Summary This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.     

Error analysis of upwind‐discretizations for the steady‐state incompressible Navier–Stokes equations

Within the framework of finite element methods, the paper investigates a general approximation technique for the nonlinear convective term of the Navier–Stokes equations. The approach is based on an upwind method of finite volume type. It is proved that the discrete convective term satisfies a well‐known collection of sufficient conditions for convergence of the finite element solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

A residual based error estimator for mortar finite element discretizations

Summary. A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the Lagrange multiplier is taken into account. Received July 18, 1997 / Revised version received July 27, 1998 / Published online September 7, 1999     

Numerical methods for reaction-diffusion problems with non-differentiable kinetics

Summary We consider a class of steady-state semilinear reaction-diffusion problems with non-differentiable kinetics. The analytical properties of these problems have received considerable attention in the literature. We take a first step in analyzing their numerical approximation. We present a finite element method and establish error bounds which are optimal for some of the problems. In addition, we also discuss a finite difference approach. Numerical experiments for one- and two-dimensional problems are reported.Dedicated to Ivo Babuka on his sixtieth birthdayResearch partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant Number AFOSR 85-0322     

Convergence and optimal complexity of adaptive finite element eigenvalue computations

In this paper, an adaptive finite element method for elliptic eigenvalue problems is studied. Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved. The analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper. This work was partially supported by the National Science Foundation of China under grant 10425105 and the National Basic Research Program under grant 2005CB321704.     

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.     

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

Summary The Lagrange-Galerkin method is a numerical technique for solving convection — dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method. We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.     

Symmetric coupling of boundary elements and Raviart–Thomas-type mixed finite elements in elastostatics

Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical examples are included. Received February 21, 1995 / Revised version received December 21, 1995     

Negative norm error control for second-kind convolution Volterra equations

Summary. We consider a piecewise constant finite element approximation to the convolution Volterra equation problem of the second kind: find such that in a time interval . An a posteriori estimate of the error measured in the norm is developed and used to provide a time step selection criterion for an adaptive solution algorithm. Numerical examples are given for problems in which is of a form typical in viscoelasticity theory. Received March 5, 1998 / Revised version received November 30, 1998 / Published online December 6, 1999