Uniformly convergent interior penalty methods using multilinear approximations for problems in elasticity

Interior penalty methods for linear elasticity, when used with bilinear elements, are effective in the context of compressible materials. However, they may produce poor approximations with these elements in the nearly incompressible regime. We propose a new general interior penalty formulation and prove that it is uniformly convergent with respect to the compressibility parameter for multilinear elements. The new formulation is a modification of well-known methods (nonsymmetric, incomplete, and symmetric interior penalty Galerkin) through underintegration of selected edge terms