the Institute for Theoretical and Engineering Science
Department of Mathematics

University of Houston

Scientific Computing Seminar

Professor Ralf Hiptmair
Seminar for Applied Mathematics, ETH Zürich

Spurious Solutions Explained

Monday, January 9, 2006$^*$
2:00 PM- 3:00 PM
Room 634 S&R1

$^*$Note Special Day and Time

Abstract: In 1988 A. Bossavit wrote a pioneering paper titled ``A rationale for edge elements'' and cited as a major argument that edge element produce ``spurious free approximations''. He referred to the fact that for certain boundary value problems in electromagnetics the use of standard continuous Lagrangian finite elements to approximate the fields will mean convergence to wrong solutions.

In my presentation I am going to shed light on the emergence of spurious solutions for a simple magnetostatic model problem. It will turn out that their existence can be blamed on the failure of the well-known identity $-\Delta = -\operatorname{\bf grad}\operatorname{div}+\operatorname{\bf curl}\operatorname{\bf curl}$ to hold in a variational context. This statement can be made rigorous using concepts from functional analysis. I am also going to discuss why edge finite elements succeed in approximating the correct solutions.

References: 1. A. BOSSAVIT, A rationale for edge elements in 3D field computations, IEEE Trans. Mag., 24 (1988), pp. 74-79. 2. M. COSTABEL AND M. DAUGE, Weighted regularization of Maxwell equations in polyhedral domains: A rehabilitation of nodal finite elements, Numer. Math., 93 (2002), pp. 239-277. 3. R. HIPTMAIR, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), pp. 237-339.

This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.