Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data

Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data

​Hall, Eric Joseph, Håkon Hoel, Mattias Sandberg, Anders Szepessy, and Raúl Tempone, "Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data", SIAM Journal on Scientific Computing, Volume 38, no. 6 (2016): A3773-A3807
Eric Joseph Hall, Håkon Hoel, Mattias Sandberg, Anders Szepessy, Raúl Tempone
a posteriori error, Galerkin error, quadrature error, elliptic PDE, random PDE, Monte Carlo methods, lognormal
2016
We derive computable error estimates for finite element approximations of linear elliptic partial differential equations with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.

1095-7197