A finite element approximation of linear stochastic PDEs driven by multiplicative white noise

References

• Molchanov , S. A. 1991 . Ideas in theory of random media . Acta Applicandae Mathematicae , 22 : 139 – 282 .
   Web of Science ®Google Scholar
• Holden , H. , Øksendal , B. , Ubøe , J. and Zhang , T. 1996 . Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach. Probability and its Applications , Basel : Birkhäuser .
   Google Scholar
• Kardar , M. and Zhang , Y. C. 1987 . Scaling of directed polymers in random media . Physical Review Letters , 58 : 2087 – 2090 .
   PubMed Web of Science ®Google Scholar
• Caraballo , T. and Robinson , J. C. 2004 . Stabilisation of linear PDEs by Stratonovich noise . Systems and Control Letters , 53 : 41 – 50 .
   Web of Science ®Google Scholar
• Caraballo , T. , Liu , K. and Mao , X. 2001 . On stabilization of partial differential equations by noise . Nagoya Math. J. , 161 : 155 – 170 .
   Web of Science ®Google Scholar
• Walsh , J. B. 1986 . “ Lecture Notes in Mathematics ” . In An Introduction to Stochastic Partial Differential Equations , Vol. 1180 , Berlin : Springer-Verlag .
   Google Scholar
• Babuška , I. and Chleboun , J. 2003 . Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions . Mathematics of Computation , 71 : 1339 – 1370 .
   Google Scholar
• Babuška , I. , Liu , K.-M. and Tempone , R. 2003 . Solving stochastic partial differential equations based on the experimental data . Mathematical Models and Methods in Applied Sciences , 13 : 415 – 444 .
   Google Scholar
• Babuška , I. , Tempone , R. and Zouraris , G. E. 2004 . Galerkin finite element approximations of stochastic elliptic partial differential equations . SIAM Journal on Numerical Analysis , 42 : 800 – 825 .
   Web of Science ®Google Scholar
• Ghanem , R. 1999 . Ingredients for a general purpose stochastic finite elements implementation . Computational Methods in Applied Mechanical Engineering , 168 : 19 – 34 .
   Web of Science ®Google Scholar
• Ghanem , R. 1999 . Stochastic finite elements for heterogeneous media with multiple random non-gaussian properties . Journal of Engineering Mechanics , 125 : 24 – 40 .
   Google Scholar
• Jardak , M. , Su , C.-H. and Karniadakis , G. E. 2002 . Spectral polynomial chaos solutions of the stochastic advection equation . SIAM Journal on Scientific Computing , 17 : 319 – 338 .
   Google Scholar
• Keese , A. and Matthies , H. G. Parallel solution of stochastic PDEs . Proceedings of Applied Mathematics and Mechanics , Vol. 2 , pp. 485 – 486 .
   Google Scholar
• Matthies , H. G. and Keese , A. Multilevel methods for stochastic systems. In: . Proceedings of the First MIT Conference . Amsterdam. Computational Fluid and Solid Mechanics , Edited by: Bathe , K.-J. pp. 1620 – 1622 . Elsevier .
   Google Scholar
• Pellissetti , M. and Ghanem , R. 2000 . Iterative solution of systems of linear equations arising in the context of stochastic finite elements . Advances in Engineering Software , 31 : 607 – 616 .
   Web of Science ®Google Scholar
• Xiu , D. and Karniadakis , G. E. 2002 . Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos . Computational Methods in Applied Mechanical Engineering , 191 : 4927 – 4948 .
   Web of Science ®Google Scholar
• Xiu , D. and Karniadakis , G. E. 2002 . Stochastic modeling of flow-structure interaction using generalized polynomial chaos . Journal of Engineering Mechanics , 124 : 51 – 69 .
   Google Scholar
• Xiu , D. , Lucor , D. , Su , C.-H. and Karniadakis , G. E. 2002 . The Wiener–Askey polynomial chaos for stochastic differential equations . SIAM Journal on Scientific Computing , 24 : 619 – 644 .
   Web of Science ®Google Scholar
• Holden , H. , Lindstrøm , T. , Øksendal , B. , Ubøe , J. and Zhang , T.-S. 1994 . The Burgers equation with a noisy force and the stochastic heat equation . Communications in Partial Differential Equations , 19 : 119 – 141 .
   Web of Science ®Google Scholar
• Holden , H. , Lindstrøm , T. , Øksendal , B. , Ubøe , J. and Zhang , T.-S. 1995 . The pressure equation for fluid flow in a stochastic medium . Potential Analysis , 4 : 655 – 674 .
   Web of Science ®Google Scholar
• Benth , F. E. and Gjerde , J. 1998 . Convergence rates for finite element approximations of stochastic partial differential equations . Stochastic and Stochastic Reports , 63 : 313 – 326 .
   Google Scholar
• Benth , F. E. and Gjerde , J. 1998 . “ Numerical solution of the pressure equation for fluid flow in a stochasticmedium. Progress in Probability ” . In Stochastic Analysis and Related Topics, VI , Vol. 42 , 175 – 186 . Basel : Birkhäuser Verlag . Geilo, 1996
   Google Scholar
• Holden , H. and Hu , Y. 1996 . Finite difference approximation of the pressure equation for fluid flow in a stochastic medium. A probabilistic approach . Communications in Partial Differential Equations , 21 : 1367 – 1388 .
   Web of Science ®Google Scholar
• Theting , T. G. 2000 . Solving Wick-stochastic boundary value problems using a finite element method . Stochastic and Stochastic Reports , 70 : 241 – 270 .
   Google Scholar
• Våge , G. 1998 . Variational methods for PDEs applied to stochastic partial differential equations . Mathematica Scandinavica , 82 : 113 – 137 .
   Web of Science ®Google Scholar
• Våge , G. 1994 . “ Hilbert space methods applied to elliptic stochastic partial differential equations ” . In Stochastic Analysis and Related Topics, V , Vol. 32 , 281 – 294 . Silviri,Basel : Birkhäuser . 1998, Progress in Probability
   Google Scholar
• Manouzi , H. and Theting , T. G. 2004 . Mixed finite element approximation for the stochastic pressure equation of Wick type . IMA Journal of Numerical Analysis , 24 : 605 – 634 .
   Google Scholar
• Hida , T. , Kuo , H. , Potthoff , J. and Streit , L. 1993 . “ Mathematics and its Applications ” . In White Noise. An infinite Dimensional Calculus , Vol. 253 , Dordrecht : Kluwer Academic Publishers .
   Google Scholar
• Filinkov , A. and Sorensen , J. 2002 . Differential equations in spaces of abstract stochastic distributions . Stochastic and Stochastic Reports , 72 : 129 – 173 .
   Google Scholar
• Gjessing , H. K. 1994 . “ Wick calculus with applications to anticipating stochastic differential equations ” . Manuscript, University of Bergen .
   Google Scholar
• Kondratiev , Y. , Leukert , P. and Striet , L. 1996 . Wick calculus in Gaussian analysis . Acta Applicandae Mathematicae , 44 : 269 – 294 .
   Web of Science ®Google Scholar
• Theting , T. G. 2003 . Solving parabolic Wick-stochastic boundary value problems using a finite element method . Stochastic and Stochastic Reports , 75 : 57 – 92 .
   Google Scholar
• Pilipovic , S. and Selesi , D. 2007 . Expansion theorems for generalized random processes, wick products and applications to infinite stochastic differential equations . Quantum Probability and Related Topics , 10 : 79 – 110 .
   Web of Science ®Google Scholar
• Haussmann , U. G. 1978 . Asymptotic stability of the linear Ito equation in infinite dimensions . Journal of Mathematical Analysis and Applications , 65 : 219 – 235 .
   Web of Science ®Google Scholar