An optimal order multigrid method for biharmonic,C 1 finite element equations

Summary In this paper, we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the Hsieh-Clough-Tocher,C ¹ macro finite elements or several otherC ¹ finite elements. Since the multipleC ¹ finite element spaces considered are not nested, the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method. This method converges with the optimal computational order.     

Steplength optimization and linear multigrid methods

Summary Recently steplength parameters have been used in linear multigrid methods. In this paper we give a theoretical analysis of the effects of steplength optimization in a rather general framework which covers two different implementations of steplength optimization in standard multigrid methods.     

A conjugate gradient method and a multigrid algorithm for Morley s finite element approximation of the biharmonic equation

Summary The numerical solution of the linear equations arising from Morley's nonconforming displacement method is studied. A suitable preconditioning for the conjugate gradient method is described. Furthermore, the nonconformity of the discretization necessitates some non-standard ingredients of multigrid algorithms.     

Estimates for multigrid methods based on red-black Gauss-Seidel smoothings

Summary TheMGR[v] algorithms of Ries, Trottenberg and Winter, the Algorithms 2.1 and 6.1 of Braess and the Algorithm 4.1 of Verfürth are all multigrid algorithms for the solution of the discrete Poisson equation (with Dirichlet boundary conditions) based on red-black Gauss-Seidel smoothing. Both Braess and Verfürth give explicit numerical upper bounds on the rate of convergence of their methods in convex polygonal domains. In this work we reconsider these problems and obtain improved estimates for theh–2h Algorithm 4.1 as well asW-cycle estimates for both schemes in non-convex polygonal domains. The proofs do not depend on the strengthened Cauchy inequality.Sponsored by the Air Force Office of Scientific Research under Contract No. AFOSR-86-0163     

A multidomain spectral collocation method for the Stokes problem

Summary We propose a multidomain spectral collocation scheme for the approximation of the two-dimensional Stokes problem. We show that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.Deceased     

On the numerical treatment of viscous flows against bodies with corners and edges by boundary element and multigrid methods

Summary The slow viscous flow past a spatial body with corners and edges is investigated mathematically and numerically by means of a boundary element method. For the resulting algebraic system a multigrid solver is designed and analyzed. Due to an improved bound on the rate of convergence it proves to be preferable to that introduced earlier for related problems. A numerical example illustrates some of the proposed methods.     

On the multi-level splitting of finite element spaces

Summary In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log )²) where is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like instead of for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.     

The frequency decomposition multi-grid method

Summary A new variant of the multi-grid algorithms is presented. It uses multiple coarse-grid corrections with particularly associated prolongations and restrictions. In this paper the robustness with respect to anisotropic problems is considered.Dedicated to the memory of Peter Henrici     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

Preconditioning indefinite discretization matrices

Summary The finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are known for these systems. We show that these preconditioning matrices can also be used for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation. Our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case.     

The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method

Summary The paper addresses the problem of the implementation of nonhomogeneous essential Dirichlet type boundary conditions in thep-version of the finite element method.Partially supported by the Office of Naval Research under Grant N-00014-85-K-0169Research partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0322     

Analysis of the Kleiser-Schumann method

Summary The Kleiser-Schumann algorithm for the approximation of the Stokes problem by Fourier/Legendre polynomials is analized. Stability when the degree of the polynomials increases is established, whereas error estimates in Sobolev spaces are proven.The research of this author has been partially supported by the U.S. Army through its European Research Office under contract No. DAJA-84-C-0035     

A class of iterative methods for solving saddle point problems

Summary We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.The work of this author was supported by the Office of Naval Research under contract N00014-82K-0197, by Avions Marcel Dassault, 78 Quai Marcel Dassault, 92214 St Cloud, France, and by Direction des Recherches Etudes et Techniques, 26 boulevard Victor, F-75996 Paris Armées, FranceThe work of this author was supported by Avions Marcel Daussault-Breguet Aviation, 78 quai Marcel Daussault, F-92214 St Cloud, France and by Direction des Recherches Etudes et Techniques, 26 boulevard Victor, F-75996 Paris Armées, FranceThe work of this author was supported by Konrad-Zuse-Zentrum für Informationstechnik Berlin, Federal Republic of Germany     

On an iterative method for variational inequalities

Summary A number of numerical solutions are presented as examples of a new iterative method for variational inequalities. The iterative method is based on the reduction of variational inequalities to the Wiener-Hopf equations. For obstacle problems the convergence is guaranteed inW ^1,p spaces forp2. The examples presented are one and two dimensional obstacle problems in cases when the Greens function is known, but the method is very general.     

Two preconditioners based on the multi-level splitting of finite element spaces

Summary 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.The author was supported by the Konrad-Zuse-Zentrum für Informationstechnik Berlin, Federal Republic of Germany     

On the convergence of the multigrid method for a hypersingular integral equation of the first kind

Summary We present a multigrid method to solve linear systems arising from Galerkin schemes for a hypersingular boundary integral equation governing three dimensional Neumann problems for the Laplacian. Our algorithm uses damped Jacobi iteration, Gauss-Seidel iteration or SOR as preand post-smoothers. If the integral equation holds on a closed, Lipschitz surface we prove convergence ofV- andW-cycles with any number of smoothing steps. If the integral equation holds on an open, Lipschitz surface (covering crack problems) we show convergence of theW-cycle. Numerical experiments are given which underline the theoretical results.     

The Neumann-Dirichlet domain decomposition method with inexact solvers on the subdomains

Summary We consider the Neumann-Dirichlet domain decomposition method for the solution of linear elliptic boundary value problems. We study the following question. Suppose that the auxiliary problems on the subdomains are not solved exactly, but only with a fixed, mesh size independent accuracy. Does the speed of convergence remain mesh size independently bounded? We show that the answer is no in general, but that mesh size independent convergence can be obtained if the accuracy requirement on the subsolvers becomes increasingly severe as the mesh size tends to zero.     

Analysis of a damped nonlinear multilevel method

Summary In this paper, we present a new algorithm that is obtained by introducing a damping parameter in the classical Nonlinear Multilevel Method. We analyse this Damped Nonlinear Multilevel Method. In particular, we prove global convergence and local efficiency for a suitable class of problems.     

A constructive method for deriving finite elements of nodal type

Summary In this paper, we propose an algorithm to derive nodal methods corresponding to various two and three-dimensional nonconforming and mixed finite elements. We show that this algorithm can be used to obtain several classical schemes as well as some more recently developed schemes, and that it leads to a simple proof of unisolvence for these methods. Finally we use our method to obtain a three dimensional nodal scheme of BDM type.     

Compactness method in the finite element theory of nonlinear elliptic problems

Summary Finite element approximation of a nonlinear elliptic pseudomonotone second-order boundary value problem in a bounded nonpolygonal domain with mixed Dirichlet-Neumann boundary conditions is studied. In the discretization we approximate the domain by a polygonal one, use linear conforming triangular elements and evaluate integrals by numerical quadratures. We prove the solvability of the discrete problem and on the basis of compactness properties of the corresponding operator (which is not monotone in general) we prove the convergence of approximate solutions to an exact weak solutionuH ¹ ). No additional assumption on the regularity of the exact solution is needed.