Gerhard Starke

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

SOR for AX−XB=C

H({\rm div};\,\Omega)^d \times H^1(\Omega)^d

A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations

norm. This immediately implies optimal error estimates for finite element subspaces of

Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis

H({\rm div};\,\Omega)^d \times H^1(\Omega)^d

FIRST-ORDER SYSTEM LEAST SQUARES FOR THE STRESS-DISPLACEMENT FORMULATION: LINEAR ELASTICITY ∗

. 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 ...

Preconditioned Krylov Subspace Methods for Lyapunov Matrix Equations

We consider a new approach to the block SOR method applied to linear systems of equations which can be written as a matrix equation AX−XB=C. Such systems arise, for example, from finite difference discretizations of seperable elliptic boundary value problems on rectangular domains. On one hand, this gives us an iterative method for the solution of such matrix equations (e.g., Lyapunovs matrix equation where B=−AT), and on the other hand, the problem of choosing appropriate parameters for the block SOR method can be written in a more compact form which may be helpful, especially, for non-self-adjoint problems, i.e., if A and B are nonsymmetric. Using this technique, we determine—under more general assumptions than those of Chin and Manteuffel—the optimal SOR parameters for the model problem of a convection-diffusion equation.

Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations

SummaryWe present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations, say of the formAx=b, whereA ∈ ℝN, N, withA nonsingular, andb ∈ ℝN are given. This hybrid method begins with a limited number of steps of the Arnoldi method to obtain some information on the location of the spectrum ofA, and then switches to a Richardson iterative method based on Faber polynomials. For a polygonal domain, the Faber polynomials can be constructed recursively from the parameters in the Schwarz-Christoffel mapping function. In four specific numerical examples of non-normal matrices, we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid GMRES or restarted versions of the GMRES algorithm. It is, however, sensitive (as other hybrid methods also are) to the amount of information on the spectrum ofA acquired during the first (Arnoldi) phase of this procedure.

A Boundary Functional for the Least-Squares Finite- Element Solution of Neutron Transport Problems

Abstract. A preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed. The proposed preconditioner is of block triangular form. The aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system. As an example, a saddle point problem obtained from a mixed finite element discretization of the Oseen equations is considered. The convergence estimates obtained by using a field–of–values analysis are independent of the discretization parameter h. Several computational experiments supplement the theoretical results and illustrate the performance of the method.

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part I

This paper develops a least-squares finite element method for linear elasticity in both two and three dimensions. The least-squares functional is based on the stress-displacement formulation with the symmetry condition of the stress tensor imposed in the first-order system. For the respective displacement and stress, using the Crouzeix--Raviart and Raviart--Thomas finite element spaces, our least-squares finite element method is shown to be optimal in the (broken) H1 and H(div) norms uniform in the incompressible limit.

First-Order System Least Squares for Coupled Stokes-Darcy Flow

The authors study the iterative solution of Lyapunov matrix equations