Super-convergence of reproducing kernel approximation

Abstract

The reproducing kernel particle methods (RKPM) are meshfree methods arising in mechanics, especially in dealing with problems involving large deformation and singularities. We provide a theoretical analysis of super-convergence in Sobolev norms for reproducing kernel (RK) approximations when the interpolation order p is even. Super-convergence phenomenon means the convergence rate is higher than the order that is generally expected. We distinguish the continuous RK approximation and the discrete RKP approximation. While the continuous RK approximations are proven to be super-convergent when p is even, its discrete counterpart has super-convergence only with uniform particle distribution and special choices of RK kernel functions and support sizes. Moreover, super-convergence does not exist for the discrete RKP approximation with general RK support sizes. The concept of pseudo-super-convergence is then introduced to explain why in practice the super-convergence phenomenon is sometimes observed for general cases although in theory it is not true. Our analysis is general for multi-dimensional RK approximations.