A finite element method for martingale-driven stochastic partial differential equations

Abstract

The main objective of this work is to describe a Galerkin ap- proximation for stochastic partial differential equations driven by square- integrable martingales. Error estimates in the semidiscrete case, where dis- cretization is only done in space, and in the fully discrete case are derived. Parabolic as well as transport equations are studied. Arguably the modern literature on Finite Elements Methods (FEM's from this point on) can be traced back to the 1956 paper of Turner, Clough, Martin and Topp (26). Several contemporary publications followed (1) (17), mostly under an Engineering scope. This should come as no surprise given the aerospace back- ground of the authors. It was not until the early 1970's that research on FEM's applied to the estimation of solutions to partial differential equations picked up steam. Here one finds, among others, the works of Fujita and Mizutani (11), Ushi- jima (27) and Zlamal (30) (31) on parabolic PDE's, together with those of Ciarlet (8) and Nedoma (19) for elliptic problems. From this point on the stream of related publications swelled in breadth and depth, from various forms of approximations (Galerkin, Riez-Galerkin, Lagrange-Galerkin, etc.) to different methodologies (energy methods, dynamic FEM's, etc.) and a more focused treatment of specific problems (notably Navier-Stokes equations). Although beyond the scope of this paper, it should be mentioned that the theory of convergence of finite-elements approximations developed in parallel fashion. Unsurprisingly, the (numerical) study of stochastic partial differential equations (SPDE's from this point on) took more time to develop. It was not until the mid- to-late 1990's that work such as that of Gyongy and Nualart (13), Yoo (29), Crisan, Gaines and Lyons (9), and Gaines (12) started breaking trail in this direction. However, none of these papers present a FEM approach. Albeit more complex to implement, FEM's do present the advantage of greater degrees of freedom when choosing a discretization of the space-time continuum. This can be particularly useful to design finite elements tailor made to the characteristics of a specific problem. To our knowledge, the 2003 paper by Yan (28) was the first to employ FEM's to estimate solutions to SPDE's, where error estimates for linear equations