Kernel-Based Collocation Methods Versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations

Abstract

We compare a kernel-based collocation method (meshfree approximation method) with a Galerkin finite element method for solving elliptic stochastic partial differential equations driven by Gaussian noises. The kernel-based collocation solution is a linear combination of reproducing kernels obtained from related differential and boundary operators centered at chosen collocation points. Its random coefficients are obtained by solving a system of linear equations with multiple random right-hand sides. The finite element solution is given as a tensor product of triangular finite elements and Lagrange polynomials defined on a finite-dimensional probability space. Its coefficients are obtained by solving several deterministic finite element problems. For the kernel-based collocation method, we directly simulate the (infinite-dimensional) Gaussian noise at the collocation points. For the finite element method, however, we need to truncate the Gaussian noise into finite-dimensional noises. According to our numerical experiments, the finite element method has the same convergence rate as the kernel-based collocation method provided the Gaussian noise is truncated using a suitable number terms.