Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients

Abstract

Dedicated to Olof B. Widlund on the occasion of his 60th birthday. SUMMARYWe prove extension theorems in the norms described by Stokes and Lamà e operators for the threedimensional case with periodic boundary conditions. For the Lamà e equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem.Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and ÿctitious domain methods, and in analysis of ÿnite element methods. We consider an application of established extension theorems to an e cient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coe cients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coe cient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning.Under some natural assumptions on smoothness and topological properties of subdomains with small coe cients, we prove convergence of the simplest Richardson method uniform in the jump of coecients. For the Lamà e equations, the convergence is also uniform in the incompressible limit.Our preliminary numerical results for two-dimensional di usion problems show fast convergence uniform in the jump and in the mesh size parameter.

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

“…However, it has been shown in the continuous case (cf. [2,3,6]) that if a special initial guess is used, then the rate of convergence does not depend on the size of the jumps even if a standard preconditioned iterative method is used. A similar result has been established in [27,4] for algebraic systems of linear equations with symmetric coefficient matrices.…”

FFT‐based homogenization for microstructures discretized by linear hexahedral elements

“…This article is based on earlier work on using fast solvers as preconditioners [34][35][36][37][38][39] and FEM on regular grids [40][41][42][43][44][45][46]. In the opinion of the authors, the derived 3D explicit expressions for the Fourier coefficients of the operators involved, the matrix-free formulation, and the treatment of nonlinear problems are innovative.…”

Observations on degenerate saddle point problems

“…We prove that the degeneracy does not affect the wellposedness in a standard norm under some natural assumptions, using ideas that are initiated by [3,4,5,6,7,14,15]. By wellposedness, contrary to illposedness, we mean a stable dependence of the solution on the right-hand side.…”

“…Degenerate saddle point problems, e.g., can be viewed as limit cases of mixed formulations of symmetric problems with large jumps in coefficients, corresponding to an infinite jump. We prove that the degeneracy does not affect the wellposedness in a standard norm under some natural assumptions, using ideas that are initiated by [3,4,5,6,7,14,15]. By wellposedness, contrary to illposedness, we mean a stable dependence of the solution on the right-hand side.…”

“…Our abstract geometrical condition of generalized wellposedness in this example is equivalent to a possibility of extension of displacements preserving the energy norm of the Lamé operator. It has been proved in [4,7] that such an extension is possible under some assumptions. We expect that in the limit case of infinitely large Lamé coefficients λ and µ in a subdomain the pseudoinverse of the operator is bounded, which makes the problem wellposed for the stresses in the L 2 sense, i.e.…”