Talks by Jan Blechta and Daniel Campbell

The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place on Wednesday from 9:00 in lecture room K3. The talks will be given by Jan Blechta and Daniel Campbell. Please see the details below.

Speaker: Jan Blechta
Title: GMRES with I+compact is superlinear. What about A+compact?

Abstract: It is known for decades that the Krylov subspace methods applied to compact perturbations of identity produce superlinear convergence and that the convergence behavior is bounded in terms of the distribution of the singular values of the perturbation. In this talk we will present a generalization of such kind of result. We consider operators that exhibit linear GMRES convergence. Such operators are precisely those that do not contain the origin in the closure of their numerical range. The main result is stability with respect to compact perturbations: If the operator is compactly perturbed, deviation from the linear convergence is controlled in terms of the singular values of the perturbation. We will give this statement a precise quantitative meaning, which allows for arbitrarily large perturbations. We will mention applications of this result and state some open questions. This result has been published as [J. Blechta, SIMAX 2021, 1137/20M1340848].

Speaker: Daniel Campbell
Title: Injectivity in second-gradient Nonlinear Elasticity

Abstract: We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that Ω ⊂ R^n is a domain, f ∈ W^(2,q)(Ω, R^n) satisfies |J_f|^(−a) ∈ L^1 and that f equals a given homeomorphism on ∂Ω. Under suitable conditions on q and a we show that f must be a homeomorphism. As a main new tool we find an optimal condition for a and q that imply that H^(n−1)({J_f = 0}) = 0 and hence J_f cannot change sign. We further specify in dependence of q and a the maximal Hausdorff dimension d of the critical set {J_f = 0}. The sharpness of our conditions for d is demonstrated by constructing respective counterexamples.