An Efficient Direct Solver for the Boundary Concentrated FEM in 2D

The boundary concentrated FEM, a variant of the hp-version of the finite element method, is proposed for the numerical treatment of elliptic boundary value problems. It is particularly suited for equations with smooth coefficients and non-smooth boundary conditions. In the two-dimensional case it is shown that the Cholesky factorization of the resulting stiffness matrix requires O(Nlog⁴ N) units of storage and can be computed with O(Nlog⁸ N) work, where N denotes the problem size. Numerical results confirm theoretical estimates. Received October 4, 2001; revised August 19, 2002 Published online: October 24, 2002     

Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients is very complicated. On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious. In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid method can be applied to solve the linear system on the fine scale. We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements (CFE–MG).     

Weighted Error Estimates for Finite Element Solutions of Variational Inequalities

In this note the studies begun in Blum and Suttmeier (1999) on adaptive finite element discretisations for nonlinear problems described by variational inequalities are continued. Similar to the concept proposed, e.g., in Becker and Rannacher (1996) for variational equalities, weighted a posteriori estimates for controlling arbitrary functionals of the discretisation error are constructed by using a duality argument. Numerical results for the obstacle problem demonstrate the derived error bounds to be reliable and, used for an adaptive grid refinement strategy, to produce economical meshes. Received September 6, 1999; revised February 8, 2000     

Stabilised hp-Finite Element Approximation of Partial Differential Equations with Nonnegative Characteristic Form

This paper is devoted to the a priori error analysis of the hp-version of a streamline-diffusion finite element method for partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic problems, first-order hyperbolic problems and second-order problems of mixed elliptic-parabolic-hyperbolic type. We derive error bounds which are simultaneously optimal in both the mesh size h and the spectral order p. Numerical examples are presented to confirm the theoretical results. Received October 28, 1999; revised May 26, 2000     

A Note on the Energy Norm for a Singularly Perturbed Model Problem

This paper considers a singularly perturbed reaction diffusion problem. It is investigated whether adaptive approaches are successful to design robust solution procedures. A key ingredient is the a posteriori error estimator. Since robust and mathematically analysed error estimation is possible in the energy norm, the focus is on this choice of norm and its implications. The numerical performance for several model problems confirms that the proposed adaptive algorithm (in conjunction with an energy norm error estimator) produces optimal results. Hence the energy norm is suitable for the purpose considered here. The investigations also provide valuable justification for forthcoming research. Received October 25, 2001; revised July 12, 2002 Published online: October 24, 2002     

Finite Element Methods for Eigenvalue Problems on a Rectangle with (Semi-) Periodic Boundary Conditions on a Pair of Adjacent Sides

This paper deals with a class of elliptic differential eigenvalue problems (EVPs) of second order on a rectangular domain Ω⊂ℝ², with periodic or semi-periodic boundary conditions (BCs) on two adjacent sides of Ω. On the remaining sides, classical Dirichlet or Robin type BCs are imposed. First, we pass to a proper variational formulation, which is shown to fit into the framework of abstract EVPs for strongly coercive, bounded and symmetric bilinear forms in Hilbert spaces. Next, the variational EVP serves as the starting point for finite element approximations. We consider finite element methods (FEMs) without and with numerical quadrature, both with triangular and with rectangular meshes. The aim of the paper is to show that well-known error estimates, established for finite element approximations of elliptic EVPs with classical BCs, remain valid for the present type of EVPs, including the case of multiple exact eigenvalues. Finally, the analysis is illustrated by a non-trivial numerical example, the exact eigenpairs of which can be determined. Received March 2, 1999; revised July 8, 1999     

ℋ-matrices for Convection-diffusion Problems with Constant Convection

L ^∞-coefficients. This paper analyses the application of hierarchical matrices to the convection-dominant convection-diffusion equation with constant convection. In the case of increasing convection, the convergence of a standard ℋ-matrix approximant towards the original matrix will deteriorate. We derive a modified partitioning and admissibility condition that ensures good convergence also for the singularly perturbed case. Received January 1, 2003; revised March 4, 2003 Published online: May 2, 2003     

Multilevel Approximation of Boundary Integral Operators

Received March 29, 2000; revised June 7, 2001     

Reliable Approximation of Weight Factors Entering Residual-based Error Bounds for FE-discretisations

In this note we refine strategies of the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes. We derive rigorous error bounds, especially we control the approximation process of the (unknown) dual solution entering the proposed estimate.     

Coupling of Nonconforming Finite Elements and Boundary Elements II: A Posteriori Estimates and Adaptive Mesh-Refinement

The coupling of nonconforming finite element and boundary element methods was established in Part I of this paper, where quasi-optimal a priori error estimates are provided. In the second part, we establish sharp a posteriori error estimates and so justify adaptive mesh-refining algorithms for the efficient numerical treatment of transmission problems with the Laplacian in unbounded domains. Received: January 26, 1998; revised February 10, 1999     

Computable Error Bounds for Coefficients Perturbation Methods

The coefficients perturbation method (CPM) is a numerical technique for solving ordinary differential equations (ODE) associated with initial or boundary conditions. The basic principle of CPM is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as the conditions associated to it. In this paper we shall develop formulae for calculating tight error bounds for CPM when this technique is applied to second order linear ODEs. Unlike results reported in the literature, ours do not require any a priori information concerning the exact error function or its derivative. The results of this paper apply in particular to the Tau Method and to any approximation procedure equivalent to it. The convergence of the derived bounds is also discussed, and illustrated numerically. Received April 5, 2002; revised June 11, 2002 Published online: December 12, 2002     

Adaptive Low-Rank Approximation of Collocation Matrices

This article deals with the solution of integral equations using collocation methods with almost linear complexity. Methods such as fast multipole, panel clustering and ℋ-matrix methods gain their efficiency from approximating the kernel function. The proposed algorithm which uses the ℋ-matrix format, in contrast, is purely algebraic and relies on a small part of the collocation matrix for its blockwise approximation by low-rank matrices. Furthermore, a new algorithm for matrix partitioning that significantly reduces the number of blocks generated is presented. Received August 15, 2002; revised September 20, 2002 Published online: March 6, 2003     

On the Convergence of a Multigrid Method for Linear Reaction-Diffusion Problems

In this note we consider discrete linear reaction-diffusion problems. For the discretization a standard conforming finite element method is used. For the approximate solution of the resulting discrete problem a multigrid method with a damped Jacobi or symmetric Gauss-Seidel smoother is applied. We analyze the convergence of the multigrid V- and W-cycle in the framework of the approximation- and smoothing property. The multigrid method is shown to be robust in the sense that the contraction number can be bounded by a constant smaller than one which does not depend on the mesh size or on the diffusion-reaction ratio. Received June 15, 2000     

A Solution Method for a Certain Nonlinear Interface Problem in Unbounded Domains

In this paper, we consider a kind of nonlinear interface problem in unbounded domains. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive the optimal error estimate of finite element approximation to the coupled FEM-BEM problem. Then we introduce a preconditioning steepest descent method for solving the discrete system by constructing a cheap domain decomposition preconditioner. Moreover, we give a complete analysis to the convergence speed of this iterative method. Received March 30, 2000; revised November 29, 2000     

Spectral Galerkin Discretization for Hydrodynamic Stability Problems

A spectral Galerkin discretization for calculating the eigenvalues of the Orr-Sommerfeld equation is presented. The matrices of the resulting generalized eigenvalue problem are sparse. A convergence analysis of the method is presented which indicates that a) no spurious eigenvalues occur and b) reliable results can only be expected under the assumption of scale resolution, i.e., that Re/p ² is small; here Re is the Reynolds number and p is the spectral order. Numerical experiments support that the assumption of scale resolution is necessary in order to obtain reliable results. Exponential convergence of the method is shown theoretically and observed numerically. Received November 11, 1998; revised March 1, 2000     

hp-adaptive Two-Level Methods for Boundary Integral Equations on Curves

p - and hp-versions of the Galerkin boundary element method for hypersingular and weakly singular integral equations of the first kind on curves. We derive a-posteriori error estimates that are based on stable two-level decompositions of enriched ansatz spaces. The Galerkin errors are estimated by inverting local projection operators that are defined on small subspaces of the second level. A p-adaptive and two hp-adaptive algorithms are defined and numerical experiments confirm their efficiency. Received August 30, 2000; revised April 3, 2001     

A variational approximation method for 2nd order elliptic eigenvalue problems in a composite structure with nonstandard boundary conditions

In this paper we deal with the finite element analysis of a class of eigenvalue problems (EVPs) in a composite structure in the plane, consisting of rectangular subdomains which enclose an intermediate region. Nonlocal boundary conditions (BCs) of Robin type are imposed on the inner boundaries, i.e. on the interfaces of the respective subdomains with the intermediate region. On the eventual interfaces between two subdomains we impose discontinuous transition conditions (TCs). Finally, we have classical local BCs at the outer boundaries. Such problems are related to some heat transfer problems e.g. in a horizontal cross section of a wall enclosing an air cave.     

Keller's Box-Scheme for the One-Dimensional Stationary Convection-Diffusion Equation

u ,∇u)=f, is to take the average onto the same mesh of the two equations of the mixed form, the conservation law div p=f and the constitutive law p=ϕ(u,∇u). In this paper, we perform the numerical analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion equation cu _x −ɛu _xx =f. In the first one, introduced by B. Courbet in [9,10], the numerical average of the diffusive flux is upwinded along the sign of the velocity, giving a first order accurate scheme. The second one is fourth order accurate. It is based onto the Euler-MacLaurin quadrature formula for the average of the diffusive flux. We emphasize in each case the link with the SUPG finite element method. Received June 7, 2001; revised October 2, 2001     

A Finite Element-Boundary Element Algorithm for Inhomogeneous Boundary Value Problems

Received January 25, 2001; revised July 17, 2001     

Nitsche Type Mortaring for some Elliptic Problem with Corner Singularities

The paper deals with Nitsche type mortaring as a finite element method (FEM) for treating non-matching meshes of triangles at the interface of some domain decomposition. The approach is applied to the Poisson equation with Dirichlet boundary conditions (as a model problem) under the aspect that the interface passes re-entrant corners of the domain. For such problems and non-matching meshes with and without local refinement near the re-entrant corner, some properties of the finite element scheme and error estimates are proved. They show that appropriate mesh grading yields convergence rates as known for the classical FEM in presence of regular solutions. Finally, a numerical example illustrates the approach and the theoretical results. Received July 5, 2001; revised February 5, 2002 Published online April 25, 2002