Heterogeneous Smoothed Finite Element Method: Convergence/Superconvergence Proof and Its Performance in High‐Contrast Composite Materials

ABSTRACT Smoothed Finite Element Method (S‐FEM) has been widely used in many engineering simulations. However, there are still a lot of theoretical problems to be solved, especially with regard to composite materials and convergence proof. Based on a novel least‐squares approximation and conservation property, we extend S‐FEM to heterogeneous materials. Firstly, the orthogonality, Softening Effect, and energy function are checked. Secondly, the interpolation error in the maximum norm is estimated in any dimension. Consequently, we get a theoretical convergence rate which has been sought since the year 2010. When restricted to one‐dimensional problems, we construct a special test function to prove the superconvergence: (1) S‐FEM flux is exact in the meaning of element‐wise integral; (2) numerical flux is exact at some point in each element; (3) physical flux can be quadratically approximated at the center of each element. At last, we present two numerical experiments: (1) conventional S‐FEM fails in high‐contrast composite materials while our new scheme performs well; (2) our flux converges quadratically.