Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise

Numerical approximation of a stochastic partial integro-differential equation driven by a space-time white noise is studied by truncating a series representation of the noise, with finite element method for spatial discretization and convolution quadrature for time discretization. Sharp-order convergence of the numerical solutions is proved up to a logarithmic factor. Numerical examples are provided to support the theoretical analysis.

[1]  Xiaobing Feng,et al.  Finite element approximations of the stochastic mean curvature flow of planar curves of graphs , 2013, 1303.5930.

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

[3]  Bangti Jin,et al.  Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data , 2016, SIAM J. Sci. Comput..

[4]  Bangti Jin,et al.  Correction of High-Order BDF Convolution Quadrature for Fractional Evolution Equations , 2017, SIAM J. Sci. Comput..

[5]  Xiaojie Wang,et al.  Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise , 2015, 1503.00073.

[6]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

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

[8]  L. Grafakos.  Classical Fourier Analysis , 2010 .

[9]  J. M. Sanz-Serna,et al.  A numerical method for a partial integro-differential equation , 1988 .

[10]  Yubin Yan,et al.  Galerkin Finite Element Methods for Stochastic Parabolic Partial Differential Equations , 2005, SIAM J. Numer. Anal..

[11]  Andreas Prohl,et al.  Computational Studies for the Stochastic Landau-Lifshitz-Gilbert Equation , 2013, SIAM J. Sci. Comput..

[12]  Buyang Li,et al.  Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise , 2017, Math. Comput..

[13]  A. Choudary,et al.  Partial Differential Equations An Introduction , 2010, 1004.2134.

[14]  C. Lubich.  Convolution quadrature and discretized operational calculus. I , 1988 .

[15]  Eduardo Cuesta,et al.  Convolution quadrature time discretization of fractional diffusion-wave equations , 2006, Math. Comput..

[16]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

[17]  Erkan Nane,et al.  Space-time fractional stochastic partial differential equations , 2014, 1409.7366.

[18]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[19]  Bangti Jin,et al.  An analysis of the Crank–Nicolson method for subdiffusion , 2016, 1607.06948.

[20]  Shing-Tung Yau,et al.  On the Schrödinger equation and the eigenvalue problem , 1983 .

[21]  Giuseppe Da Prato,et al.  Some results on stochastic convolutions arising in Volterra equations perturbed by noise , 1996 .

[22]  C. Lubich.  Convolution quadrature and discretized operational calculus. II , 1988 .

[23]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[24]  Jacques Printems,et al.  Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation , 2012, Math. Comput..

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

[26]  Julia,et al.  Vector-valued Laplace Transforms and Cauchy Problems , 2011 .

[27]  Vidar Thomée,et al.  Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term , 1996, Math. Comput..

[28]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[29]  Ari Laptev,et al.  Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces , 1997 .