Overcoming Element-Shape Dependence of Finite Elements with Adaptive Extended Stencil FEM

The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. The Adaptive Extended Stencil Finite Element Method (AES-FEM) overcomes this dependence on element shape quality. AES-FEM replaces the traditional basis functions with basis functions constructed using local weighted least-squares approximations. AES-FEM can use higher-degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. The numerical results demonstrate that AES-FEM is more accurate than the classical FEM, is also more efficient in terms of the time-to-error ratio, and enables much better stability and faster convergence of iterative solvers over poor-quality meshes.