Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

Abstract We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L2 projection operator. Optimal strong convergence error estimates in the L2 and $\dot{H}^{-1}$ norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case.

[1]  J. Zabczyk.  Parabolic equations on Hilbert spaces , 1999 .

[2]  Zhimin Zhang,et al.  Finite element and difference approximation of some linear stochastic partial differential equations , 1998 .

[3]  P. Kloeden,et al.  Time-discretised Galerkin approximations of parabolic stochastic PDE's , 1996, Bulletin of the Australian Mathematical Society.

[4]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[5]  Yubin Yan,et al.  Error Analysis and Smoothing Properties of Discretized Deterministic and Stochastic Parabolic Problems , 2003 .

[6]  Qiang Du,et al.  Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises , 2002, SIAM J. Numer. Anal..

[7]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[8]  J. Printems.  On the discretization in time of parabolic stochastic partial differential equations , 2001 .

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

[10]  Konstantinos Chrysafinos,et al.  Error Estimates for Semidiscrete Finite Element Approximations of Linear and Semilinear Parabolic Equations Under Minimal Regularity Assumptions , 2002, SIAM J. Numer. Anal..

[11]  P. Kloeden,et al.  LINEAR-IMPLICIT STRONG SCHEMES FOR ITO-GALKERIN APPROXIMATIONS OF STOCHASTIC PDES , 2001 .

[12]  Fausto Gozzi,et al.  Regularity of solutions of a second order hamilton-jacobi equation and application to a control problem , 1995 .

[13]  R. Curtain,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS IN HILBERT SPACE , 1971 .

[14]  Law equivalence of solutions of some linear stochastic equations in Hilbert spaces , 1992 .

[15]  Jerzy Zabczyk,et al.  Stochastic evolution equations with a spatially homogeneous Wiener process , 1997 .

[16]  D. Dawson.  Stochastic evolution equations , 1972 .

[17]  E. Hausenblas.  Numerical analysis of semilinear stochastic evolution equations in Banach spaces , 2002 .

[18]  E. Hausenblas.  Approximation for Semilinear Stochastic Evolution Equations , 2003 .

[19]  I. Gyöngy.  Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White Noise I , 1998 .

[20]  T. G. Theting,et al.  Solving wick-stochastic boundary value problems using a finite element method , 2000 .

[21]  D. Nualart,et al.  Implicit Scheme for Stochastic Parabolic Partial Diferential Equations Driven by Space-Time White Noise , 1997 .

[22]  T. G. Theting,et al.  Solving parabolic Wick-stochastic boundary value problems using a finite element method , 2003 .

[23]  Giuseppe Da Prato,et al.  Maximal regularity for stochastic convolutions in \( L^p \) spaces , 1998 .

[24]  Jessica G. Gaines,et al.  Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations , 2001, Math. Comput..

[25]  T. Shardlow.  Numerical methods for stochastic parabolic PDEs , 1999 .

[26]  G. Lord,et al.  A numerical scheme for stochastic PDEs with Gevrey regularity , 2004 .