Error and Stability Estimates of a Least-Squares Variational Kernel-Based Method for Second Order Elliptic PDEs

TL;DR

This paper introduces a novel least-squares variational kernel-based method for second order elliptic PDEs that produces symmetric positive definite systems without requiring self-adjointness or boundary subspace conditions, supported by error analysis and computational experiments.

Contribution

It presents a new kernel-based scheme for elliptic PDEs that ensures positive definiteness and symmetry without boundary subspace constraints, with comprehensive error analysis.

Findings

Method yields symmetric positive definite systems.

Error bounds supported by computational experiments.

Condition number relates to kernel smoothness and discretization.

Abstract

We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is self-adjoint or positive definite as it should be in the Rayleigh-Ritz setting. However, the new scheme leads to a symmetric and positive definite algebraic system of equations. Moreover, the resulting method does not rely on certain subspaces satisfying the boundary conditions. The trial space for discretization is provided via standard kernels that reproduce the Sobolev spaces as their native spaces. The error analysis of the method is given, but it is partly subjected to an inverse inequality on the boundary which is still an open problem. The condition number of the final linear system is approximated in terms of the smoothness of the kernel and the discretization quality. Finally, the results of some computational experiments support the theoretical error bounds.