Speaker: Mark Ainsworth

Abstract

Adaptive finite element and boundary element schemes generally give rise to non-quasiuniform meshes where the ratio of the diameter of the largest to the smallest element in the mesh grows exponentially with the number of adaptive refinement steps. We study the effect on the condition number of the stiffness matrix due to this large global mesh ratio and show that it typically leads to an exponential growth in the condition number. This naturally leads to the question of how one may counteract this effect at minimal cost and to this end, we consider the effect of applying simple point relaxation smoothers and show that, roughly speaking, these serve to remove the ill-effects of a large global mesh ratio for standard nodal finite element bases.

However, this result does not extend to the situation where one uses edge or face finite elements typically employed for the discretisation of problems posed in H(div) and H(curl). We study the conditioning of the resulting stiffness matrices and again show that the matrices are generally rather poorly conditioned due to the large global mesh ratio, and show that simple point relaxation schemes cannot remove this effect. We trace the reason behind this difference and give a simple remedy for curing the problem.

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