The Least Squares Finite Element Method for Elasticity Interface Problem on Unfitted Mesh

Fanyi Yang

paper

The Least Squares Finite Element Method for Elasticity Interface Problem on Unfitted Mesh

arXiv:2306.08801

Abstract

In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is $C^{2}$ or polygonal, and the exact solution $(σ, u)$ belongs to $H^{s} (d i v; Ω_{0} \cup Ω_{1}) \times$ H^{1+s}(Ω_0 \cup Ω_1) $w i t h$ s > 1/2 $. T w o t y p eso f l e a s t s q u a r es f u n c t i o na l s a r e d e f in e d t osee k t h e n u m er i c a l so l u t i o n . T h e f i r s t i s d e f in e d b y s im pl y a ppl y in g t h e$ L^2 $n or m l e a s t s q u a r es p r in c i pl e, an d r e q u i r es t h eco n d i t i o n$ s \geq 1 $. T h eseco n d i s d e f in e d w i t ha d i scr e t e min u s n or m, w hi c hi sr e l a t e d t o t h e inn er p r o d u c t in$ H^{-1/2}(Γ) $. T h e u seo f t hi s d i scr e t e min u s n or m r es u l t s inam e t h o d o f o pt ima l co n v er g e n cer a t es an d a l l o w s t h ee x a c t so l u t i o nha s t h er e g u l a r i t y o f an y$ s > 1/2 $. T h es t abi l i t y n e a r t h e in t er f a ce f or b o t hm e t h o d s i s g u a r an t ee d b y t h e g h os tp e na l t y bi l in e a r f or m s an d w ec an d er i v e t h er o b u s t co n d i t i o nn u mb er es t ima t es . T h eco n v er g e n cer a t es u n d er$ L^2$ norm and the energy norm are derived for both methods. We illustrate the accuracy and the robustness of the proposed methods by a series of numerical experiments for test problems in two and three dimensions.