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

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 89)

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.

Keywords

Kernel-based collocation Meshfree approximation Galerkin finite element Elliptic stochastic partial differential equations Gaussian fields Reproducing kernels 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA

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