SoftFEM for spectral approximation of an second-order elliptic operator


The spectral approximation of differential operators is one of the most challenging problems in scientific computing. In this talk, we will first introduce the Galerkin formulation for the numerical spectral approximation of an second-order elliptic operator, followed by the introduction of a novel method which we referred to as softFEM. The main idea of softFEM is to reduce the stiffness of the variational problem by subtracting to the standard stiffness bilinear form a least-squares penalty on the gradient jumps across the mesh interfaces. We will briefly discuss the motivation and why one wants to soften the stiffness of the resulting systems arising from the classical FEM. We also discuss its connection to the Continuous Interior Penalty Finite Element Method. On the analysis, we give a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form and then prove that softFEM delivers the optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. We show some numerical examples for the second- and fourth-order elliptic operators to demonstrate the performance of the method. Lastly, we extend the idea to soften the stiffness of the discretised systems for the isogeometric elements and report briefly our recent results. This is a joint work with Alexandre Ern.


Quanling Deng is a Lecturer at the ANU School of Computing. He was born in Hunan, China and moved to the USA to study mathematics in August 2011. He graduated with a Ph.D. in computational mathematics with a topic on finite element analysis at the University of Wyoming in May 2016. He then joined Curtin University in Australia as a research associate and mainly contributed to the development of isogeometric analysis. He was a short-term visiting scholar at INRIA Paris, AGH University of Science and Technology in Poland, École des Ponts ParisTech (ENPC), USTC, and others. In March 2020, he joined the Department of Mathematics at the University of Wisconsin-Madison as a Van Vleck visiting assistant professor and worked on modelling and prediction of Arctic sea-ice dynamics. He joined ANU in February 2022.

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