Mass lumping and numerical quadrature preserve the optimal Galerkin convergence rates for covariance functions of fractional elliptic SPDEs, with rates determined by the fractional power, dimension, and multiplier regularity.
How to solve the stochastic partial differential equation that gives a Mat\'ern random field using the finite element method
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abstract
This tutorial teaches parts of the finite element method (FEM), and solves a stochastic partial differential equation (SPDE). The contents herein are considered "known" in the numerics literature, but for statisticians it is very difficult to find a resource for learning these ideas in a timely manner (without doing a year's worth of courses in numerics). The goal of this tutorial is to be pedagogical and explain the computations/theory to a statistician. This is not a practical tutorial, there is little computer code, and no data analysis.
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Mass Lumping and Numerical Quadrature for Approximation of Fractional Elliptic Differential Equations Driven by Gaussian White Noise
Mass lumping and numerical quadrature preserve the optimal Galerkin convergence rates for covariance functions of fractional elliptic SPDEs, with rates determined by the fractional power, dimension, and multiplier regularity.