Finite Element and Discontinuous Galerkin Method for Stochastic Helmholtz Equation in Two- and Three-dimensions

Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using these so-called stochastic partial differential equations (SPDEs) is that SPDEs are able to more fully capture the behavior of interesting phenomena; it also means that the corresponding numerical analysis of the model will require new tools to model the systems, produce the solutions, and analyze the information stored within the solutions. In the last decade, many researchers have studied different SPDEs and various numerical methods and approximation schemes for SPDEs have also been developed, analyzed, and tested [1, 4, 7, 8, 9, 10, 12, 13, 22]. In [4, 12], the analysis based on the traditional finite element method was successfully used on partial differential equations with random coefficients, using the tensor product between the deterministic and random variable spaces. Numerical methods for SPDEs with random forcing terms have also been studied in [7, 9]. In this paper, we study the following stochastic Helmholtz equation driven by an additive white noise forcing term: