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

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 $(\sigma ,u)$ belongs to $H^s(div;{\mathrm{\Omega }}_0\cup {\mathrm{\Omega }}_1)\times$H^{1+s}(Ω_0 \cup Ω_1)$with$s > 1/2$\mathrm{.}Twotypesofleastsquaresfunctionalsaredefinedtoseekthenumericalsolution\mathrm{.}Thefirstisdefinedbysimplyapplyingthe$L^2$normleastsquaresprinciple,andrequiresthecondition$s \geq 1$\mathrm{.}Thesecondisdefinedwithadiscreteminusnorm,whichisrelatedtotheinnerproductin$H^{-1/2}(Γ)$\mathrm{.}Theuseofthisdiscreteminusnormresultsinamethodofoptimalconvergenceratesandallowstheexactsolutionhastheregularityofany$s > 1/2$\mathrm{.}Thestabilityneartheinterfaceforbothmethodsisguaranteedbytheghostpenaltybilinearformsandwecanderivetherobustconditionnumberestimates\mathrm{.}Theconvergenceratesunder$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.