Stochastic partial differential equations (SPDEs) of evolutionary type are used to model continuous-time random dynamics in infinite-dimensional state spaces. They have been an active area of research for a number of decades and a deep well-developed theory is now available. A comparable numerical analysis of such equations was delayed by various technical difficulties, but there has been considerable progress in recent years.
The NSF/CBMS Regional Conference in the Mathematical Sciences on Recent Advances in the Numerical Approximation of Stochastic Partial Differential Equations, which was held at the Department of Mathematics, Illinois Institute of Technology, Chicago, in August 2010, was a catalyst for this special issue, although it is not a proceedings of that conference. We thank Abdul Khaliq, an Editor in Chief of the journal, for his encouragement to prepare this special issue.
The aim of the special issue is to provide readers with a representative selection of papers, ranging from the theoretical to the quite practical, on the these developments. It contains nine papers:
| • | Petru A. Cioica and Stephan Dahlke, Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains Citation6, | ||||
| • | Sonja Cox and Erika Hausenblas, Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise Citation7, | ||||
| • | Andreas Barth and Annika Lang, Multilevel Monte Carlo method with applications to stochastic partial differential equations Citation2, | ||||
| • | Dirk Blömker, Wael W. Mohammed, Christian Nolde and Franz Wöhrl, Numerical study of amplitude equations for SPDEs with degenerate forcing Citation3, | ||||
| • | Erich Carelli, Alexander Müller and Adreas Prohl, Domain decomposition strategies for the stochastic heat equation Citation4, | ||||
| • | Igor Cialenco, Gregory E. Fasshauer and Qi Ye, Approximation of stochastic partial differential equations by a kernel-based collocation method Citation5, | ||||
| • | Christoph Reisinger, Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative Citation9, | ||||
| • | Mofdi El-Amrani, Mohammed Seaid and Mostafa Zahri, A stabilized finite element method for stochastic incompressible Navier–Stokes equations Citation8, | ||||
| • | Iyabo A. Adamu and Gabriel J. Lord, Numerical approximation of multiplicative SPDEs Citation1. | ||||
Workshops on the numerical approximation of stochastic PDEs and related topics are now being held regularly in Europe and North America, and we expect many more to take place in the future as the methods become more widely known and used in the scientific community. This, in turn, will stimulate further mathematical developments.
References
- Adamu , I. A. and Lord , G. J. 2012 . Numerical approximation of multiplicative SPDEs . Int. J. Comput. Math. , 89 ( 18 ) : 2603 – 2621 .
- Barth , A. and Lang , A. 2012 . Multilevel Monte Carlo method with applications to stochastic partial differential equations . Int. J. Comput. Math. , 89 ( 18 ) : 2479 – 2498 .
- Blömker , D. , Mohammed , W. W. , Nolde , C. and Wöhrl , F. 2012 . Numerical study of amplitude equations for SPDEs with degenerate forcing . Int. J. Comput. Math. , 89 ( 18 ) : 2499 – 2516 .
- Carelli , E. , Müller , A. and Prohl , A. 2012 . Domain decomposition strategies for the stochastic heat equation . Int. J. Comput. Math. , 89 ( 18 ) : 2517 – 2542 .
- Cialenco , I. , Fasshauer , G. E. and Ye , Q. 2012 . Approximation of stochastic partial differential equations by a kernel-based collocation method . Int. J. Comput. Math. , 89 ( 18 ) : 2543 – 2561 .
- Cioica , P. A. and Dahlke , S. 2012 . Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains . Int. J. Comput. Math. , 89 ( 18 ) : 2443 – 2459 .
- Cox , S. and Hausenblas , E. 2012 . Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise . Int. J. Comput. Math. , 89 ( 18 ) : 2460 – 2478 .
- El-Amrani , M. , Seaid , M. and Zahri , M. 2012 . A stabilized finite element method for stochastic incompressible Navier-Stokes equations . Int. J. Comput. Math. , 89 ( 18 ) : 2576 – 2602 .
- Reisinger , C. 2012 . Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative . Int. J. Comput. Math. , 89 ( 18 ) : 2562 – 2575 .