A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under weak regularity assumptions

Carsten Carstensen,Wolfgang Boiger

doi:10.1186/2213-7467-1-5

Abstract

The discretisation of degenerate convex minimisation problems experiences numerical difficulties with a singular or nearly singular Hessian matrix. Some discrete analog of the surface energy in microstrucures is added to the energy functional to define a stabilisation technique. This paper proves (a) strong convergence of the stress even without any smoothness assumption for a class of stabilised degenerate convex minimisation problems. Given the limitted a priori error control in those cases, the sharp a posteriori error control is of even higher relevance. This paper derives (b) guaranteed a posteriori error control via some equilibration technique which does not rely on the strict Galerkin orthogonality of the unperturbed problem. In the presence of L2 control in the original minimisation problem, some realistic model scenario with piecewise smooth exact solution allows for strong convergence of the gradients plus refined a posteriori error estimates. This paper presents (c) an improved a posteriori error control in this interface problem and so narrows the efficiency reliability gap. Numerical experiments illustrate the theoretical convergence rates for uniform and adaptive mesh-refinements and the improved a posteriori error control for four benchmark examples in the computational microstructures.

Highlights

The discretisation of degenerate convex minimisation problems experiences numerical difficulties with a singular or nearly singular Hessian matrix
This paper studies the stabilisation technique of [12] and addresses the question of convergence (i) without extra regularity assumptions, (ii) in a realistic scenario called model interface problem, and (iii) establishes an a posteriori error control
Effects of stabilisation The empirical convergence rates of the error estimators ηF, ηR and the errors u − u L2( ) and σ − σ Lp ( ) for uniform mesh-refinement with and without stabilisation coincide. This indicates that the choice γ = 1 leads to some significant perturbation but maintains the correct convergence rate at the same time

Summary

Introduction

The discretisation of degenerate convex minimisation problems experiences numerical difficulties with a singular or nearly singular Hessian matrix. Infimising sequences of variational problems with non-quasiconvex energy densities, in general, develop finer and finer oscillations with no classical limit in Sobolev function spaces called microstructure [1,2,3,4,5,6]. Those oscillations cause difficulty to numerical methods because fine grids are necessary to resolve such oscillations which results in ineffective and tricky mesh-depending computations. Applications of relaxation techniques include models in computational microstructure [5,6,7], some optimal design problems [8,9], the nonlinear Laplacian [10] (where the Hessian can become arbitrarily ill-conditioned in spite of its strict convexity) and elastoplasticity [1]

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Discussion

Conclusion