Finite element approximation of stochastic partial differential equations driven by poisson random measures of jump type

SIAM Journal on Numerical Analysis

Volumn 46, Issue 1, 2007, Pages 437-471

  Hausenblas, Erika ^a  

^a UNIVERSITY OF SALZBURG   (Austria)

Author keywords

Numerical approximation; Poisson random measure; Space discretization; Stochastic evolution equations; Stochastic partial differential equations; Time discretization

Indexed keywords

COMPUTATIONAL FLUID DYNAMICS; DIFFERENTIAL EQUATIONS; IMAGE SEGMENTATION; PARTIAL DIFFERENTIAL EQUATIONS; POISSON DISTRIBUTION; POISSON EQUATION; RANDOM PROCESSES;

NUMERICAL APPROXIMATION; POISSON RANDOM MEASURE; SPACE DISCRETIZATION; STOCHASTIC EVOLUTION EQUATIONS; STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS; TIME DISCRETIZATION;

POLYNOMIAL APPROXIMATION;

EID: 55549131924     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/050654141     Document Type: Article

References (44)

1. E. J. ALLEN, S. J. NOVOSEL, AND Z. ZHANG, Finite element and difference approximation of some linear stochastic partial differential equations, Stoch. Stoch. Rep., 64 (1998), pp. 117-142.
2. D. APPLEBAUM AND J.-L. WU, Stochastic partial differential equations driven by Ĺevy space-time white noise, Random Oper. Stochastic Equations, 8 (2000), pp. 245-259.
3. S. ALBEVERIO, J. WU, AND T. ZHANG, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl., 74 (1998), pp. 21-36.
4. J. BERGH AND J. LÖFSTRÖM, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer-Verlag, New York, 1976.
5. S. BRENNER AND R. SCOTT, The Mathematical Theory of Finite Element Methods, 2nd ed., Texts Appl. Math. 15, Springer-Verlag, New York, 2002.
6. C. CARDON-WEBER AND A. MILLET, On strongly Petrovskiǐ's parabolic SPDEs in arbitrary dimension and application to the stochastic Cahn-Hilliard equation, J. Theoret. Probab., 17 (2004), pp. 1-49.
7. AL. ERN AND J. GUERMOND, Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer-Verlag, New York, 2004.
8. H. FUJITA, N. SAITO, AND T. SUZUKI, Operator Theory and Numerical Methods, Stud. Math. Appl. 30, North-Holland, Amsterdam, 2001.
9. A. GARDÓN, The order of approximations for solutions of Itô-type stochastic differential equations with jumps, Stochastic Anal. Appl., 22 (2004), pp. 679-699.
10. I. GYÖNGY, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I, Potential Anal., 9 (1998), pp. 1-25.
11. I. GYÖNGY, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II, Potential Anal., 11 (1999), pp. 1-37.
12. I. GYÖNGY AND A. MILLET, On discretization schemes for stochastic evolution equations, Potential Anal., 23 (2005), pp. 99-134.
13. d by a finite-element technique, Stochastics, 23 (1988), pp. 131-148.
14. E. HAUSENBLAS, Numerical analysis of semilinear stochastic evolution equations in Banach spaces, J. Comput. Appl. Math., 147 (2002), pp. 485-516.
15. E. HAUSENBLAS, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), pp. 141-186.
16. E. HAUSENBLAS, Existence, uniqueness, and regularity of parabolic SPDEs driven by Poisson random measure, Electron. J. Probab., 10 (2005), pp. 1496-1546.
17. E. HAUSENBLAS, SPDEs driven by Poisson random measure with non-Lipschitz coefficients, Probab. Theory Related Fields, 137 (2007), pp. 161-200.
18. E. HAUSENBLAS AND I. MARCHIS, The numerical approximation of parabolic stochastic partial differential equations driven by Poisson random measure, BIT, 46 (2006), pp. 773-811.
19. E. HAUSENBLAS AND I. MARCHIS, Numerical Simulations of Parabolic Stochastic Partial Differential Equations Driven by Poisson Random Measure, manuscript, 2006.
20. D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981.
21. G. KALLIANPUR AND J. XIONG, Diffusion approximation of nuclear space-valued stochastic-differential equations driven by Poisson random measures, Ann. Appl. Probab., 5 (1995), pp. 493-517.
22. M. KOURITZIN AND H. LONG, Convergence of Markov chain approximations to stochastic reaction-diffusion equations, Ann. Appl. Probab., 12 (2002), pp. 1039-1070.
23. M. KOURITZIN, H. LONG, AND W. SUN, On Markov chain approximations to semilinear partial differential equations driven by Poisson measure noise, Stochastic Anal. Appl., 21 (2003), pp. 419-441.
24. P. E. KLOEDEN AND S. SHOTT, Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs, J. Appl. Math. Stoch. Anal., 14 (2001), pp. 47-53.
25. D. F. KUZNETSOV, New representations of explicit one-step numerical methods for stochastic differential equations with a jump component, Zh. Vychisl. Mat. Mat. Fiz., 41 (2001), pp. 922-937.
26. G. KALLIANPUR AND J. XIONG, A nuclear space-valued stochastic differential equation driven by Poisson random measures, in Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Inform. Sci. 176, Springer-Verlag, New York, 1987, pp. 135-143.
27. G. KALLIANPUR AND J. XIONG, Stochastic Differential Equations in Infinite Dimensional Spaces, Lecture Notes Monograph Series 26, Institute of Mathematical Statistics, Hayward, CA, 1996.
28. W. LINDE, Probability in Banach Spaces - Stable and Infinitely Divisible Distributions, 2nd ed., Wiley-Interscience, New York, 1986.
29. A. MILLET AND P. MORIEN, On implicit and explicit discretization schemes for parabolic SPDEs in any dimension, Stochastic Process. Appl., 115 (2005), pp. 1073-1106.
30. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983.
31. S. PESZAT AND J. ZABCZYK, Stochastic Partial Differential Equations with Ĺevy Noise, Encyclopedia Math. Appl. 113, Cambridge University Press, Cambridge, 2007.
32. J. PRINTEMS, On the discretization in time of parabolic stochastic partial differential equations, Monte Carlo Methods Appl., 7 (2001), pp. 359-368.
33. J. PRINTEMS, On the discretization in time of parabolic stochastic partial differential equations, M2AN Math. Model. Numer. Anal., 35 (2001), pp. 1055-1078.
34. P. PROTTER, Stochastic Integration and Differential Equations, 2nd ed., Appl. Math. 21, Springer-Verlag, New York, 2004.
35. P. PROTTER AND D. TALAY, The Euler scheme for Ĺevy driven stochastic differential equations, Ann. Probab., 25 (1997), pp. 393-423.
36. S. RUBENTHALER, Numerical simulation of the solution of a stochastic differential equation driven by a Ĺevy process, Stochastic Process. Appl., 103 (2003), pp. 311-349.
37. T. RUNST AND W. SICKEL, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, de Gruyter Ser. Nonlinear Anal. Appl. 3, de Gruyter, Berlin, 1996.
38. E. ST. L. BÍE, ́Etude d'une EDPS conduite par un bruit Poissonnien. (Study of a stochastic partial differential equation driven by a Poisson noise), Probab. Theory Related Fields, 111 (1998), pp. 287-321.
39. T. SHARDLOW, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim., 20 (1999), pp. 121-145.
40. J. WALSH, An introduction to stochastic partial differential equations, in Ecole d'Et́e de Probabilit́es de Saint-Flour XIV - 1984, Lecture Notes in Math. 1180, Springer-Verlag, New York, 1986, pp. 265-437.
41. J. B. WALSH, Finite element methods for parabolic stochastic PDE's, Potential Anal., 23 (2005), pp. 1-43.
42. W. A. WOYCZÝNSKI, Ĺevy Processes in the Physical Sciences, Ĺevy Processes, Birkhäuser Boston, Boston, MA, 2001, pp. 241-266.
43. Y. YAN, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., 43 (2005), pp. 1363-1384.
44. 1 by a finite-difference method, Math. Comp., 69 (2000), pp. 653-666.