Optimal error estimates of least-squares variational kernel-based methods for second-order PDEs

Salar Seyednazari, Mehdi Tatari, Davoud Mirzaei

Published 2018-02-16Version 1

We consider meshfree least-squares variational principles to the numerical solution of partial differential equations. Indeed, in this work we focus on the development of least-squares principles for meshfree methods to find numerical solution of general second order ADN elliptic boundary value problems in $\Omega \subset \mathbb{R}^d$ under Dirichlet boundary conditions arising in engineering fields. Most notably, in these principles it is not assumed that differential operator is self-adjoint or positive definite as it would have to be in the Rayleigh-Ritz setting. Furthermore, the use of our scheme leads to symmetric and positive definite algebraic problems and allows us to circumvent compatibility conditions arising in standard and mixed-Galerkin methods. In particular, the resulting method does not require subspaces satisfying any boundary conditions. It will be illustrated how the method applies to second order elliptic systems occurring in practice with kernels that reproduce $H^\tau(\Omega)$, and the error estimates based on Sobolev norms are derived by the authors. For weighted discrete least-squares principles, it is shown that the approximations to the solutions of the second order linear elliptic boundary value problems are in optimal rate. The results of some computational experiments are also provided.