Partial Differential Equations with Random Input Data: A Perturbation Approach

Abstract

We study the numerical approximation of partial differential equations with random input data. Such problems arise when the uncertainty of the underlying system is taken into account using a probability setting. The main goal of this paper is to review the perturbation approach used in [1] for the random space approximation. The idea of this technique is to expand the exact random solution in power series of a (small) parameter $$\varepsilon$$ that characterizes the amount of uncertainty of the problem. This method yields deterministic problems that are decoupled for the coefficients building the term of a fixed power of $$\varepsilon$$ . Each problem can then be solved approximately using standard methods, such as the finite element method for the physical space discretization and an Euler scheme for time integration, as considered here. We apply the proposed methodology to several different problems, starting with an elliptic model problem with a random coefficient to facilitate the presentation. For each problem, focus is made on the derivation of (residual-based) a posteriori error estimates that take the various sources of error into account.