Solving wick-stochastic boundary value problems using a finite element method

A class of linear elliptic Wick-stochastic boundary value problems is considered. The problems are formulated in a variational (or weak) form and existence and uniqueness of a solution to this variational formulation is proved under general assumptions on the data. Furthermore, a Galerkin type of finite element method is formulated and presented in an algorithmic form. As an illustration, the algorithm is then applied to the Wick-stochastic pressure equation in one and two dimensions.

[1]  Holden Helge,et al.  Finite difference approximation of the pressure equation for fluid flow in a stochastic medium — a probabilistic approach , 1996 .

[2]  Terence Chan.  Scaling Limits of Wick Ordered KPZ Equation , 2000 .

[3]  L. Streit,et al.  Spaces of White Noise distributions: constructions, descriptions, applications. I☆ , 1993 .

[4]  F. Benth,et al.  Numerical Solution of the Pressure Equation for Fluid Flow in a Stochastic Medium , 1998 .

[5]  Bernt Øksendal,et al.  The Burgers equation with a noisy force and the stochastic heat equation , 1994 .

[6]  Yuri Kondratiev,et al.  Wick calculus in Gaussian analysis , 1996, Acta Applicandae Mathematicae.

[7]  Hans Petter Langtangen,et al.  Computational Partial Differential Equations - Numerical Methods and Diffpack Programming , 1999, Lecture Notes in Computational Science and Engineering.

[8]  B. Øksendal,et al.  THE STOCHASTIC WICK-TYPE BURGERS EQUATION , 1995 .

[9]  B. Øksendal,et al.  AN EQUATION MODELLING TRANSPORT OF A SUBSTANCE IN A STOCHASTIC MEDIUM , 1995 .

[10]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[11]  G. Våge.  Hilbert Space Methods Applied to Elliptic Stochastic Partial Differential Equations , 1996 .

[12]  Bernt Øksendal,et al.  Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach , 1996 .

[13]  Gjermund Våge,et al.  Variational methods for PDEs aplied to stochastic partial differential equations , 1998 .

[14]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[15]  Jon Gjerde.  The Pressure Equation in Anisotropic Medium , 1996 .

[16]  F. Benth,et al.  Convergence Rates for Finite Element Approximations of Stochastic Partial Differential Equations , 1998 .

[17]  Fred Espen Benth,et al.  SOME REGULARITY RESULTS FOR THE STOCHASTIC PRESSURE EQUATION OF WICK-TYPE , 2002 .

[18]  Jürgen Potthoff,et al.  White Noise: An Infinite Dimensional Calculus , 1993 .

[19]  B. Øksendal,et al.  The pressure equation for fluid flow in a stochastic medium , 1994 .