Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise
This paper aims to investigate the finite element weak convergence rate for semilinear parabolic stochastic partial differential equations(SPDEs) driven by additive noise. In contrast to many results in the current scientific literature, we investigate the more general case where the nonlinearity is...
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| Format: | Article |
| Language: | English |
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Elsevier
2023-02-01
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| Series: | Results in Applied Mathematics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2590037422000747 |
| _version_ | 1797895528570159104 |
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| author | Antoine Tambue Jean Daniel Mukam |
| author_facet | Antoine Tambue Jean Daniel Mukam |
| author_sort | Antoine Tambue |
| collection | DOAJ |
| description | This paper aims to investigate the finite element weak convergence rate for semilinear parabolic stochastic partial differential equations(SPDEs) driven by additive noise. In contrast to many results in the current scientific literature, we investigate the more general case where the nonlinearity is allowed to be of Nemytskii-type and the linear operator is not necessarily self-adjoint, which is more challenging and models more realistic phenomena such as convection–reaction–diffusion processes. Using Malliavin calculus, Kolmogorov equations and by splitting the linear operator into a self-adjoint and non self-adjoint parts, we prove the convergence of the finite element approximation and obtain a weak convergence rate that is twice the strong convergence rate. |
| first_indexed | 2024-04-10T07:27:17Z |
| format | Article |
| id | doaj.art-298e422ac1d7425f91315bc2fb651e8f |
| institution | Directory Open Access Journal |
| issn | 2590-0374 |
| language | English |
| last_indexed | 2024-04-10T07:27:17Z |
| publishDate | 2023-02-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Results in Applied Mathematics |
| spelling | doaj.art-298e422ac1d7425f91315bc2fb651e8f2023-02-24T04:31:32ZengElsevierResults in Applied Mathematics2590-03742023-02-0117100351Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noiseAntoine Tambue0Jean Daniel Mukam1Department of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, Inndalsveien 28, 5063 Bergen, Norway; Corresponding author.Department of Mathematics, Bielefeld University, 33501 Bielefeld, GermanyThis paper aims to investigate the finite element weak convergence rate for semilinear parabolic stochastic partial differential equations(SPDEs) driven by additive noise. In contrast to many results in the current scientific literature, we investigate the more general case where the nonlinearity is allowed to be of Nemytskii-type and the linear operator is not necessarily self-adjoint, which is more challenging and models more realistic phenomena such as convection–reaction–diffusion processes. Using Malliavin calculus, Kolmogorov equations and by splitting the linear operator into a self-adjoint and non self-adjoint parts, we prove the convergence of the finite element approximation and obtain a weak convergence rate that is twice the strong convergence rate.http://www.sciencedirect.com/science/article/pii/S2590037422000747Semilinear parabolic partial differential equationsFinite element methodWeak convergenceAdditive noise |
| spellingShingle | Antoine Tambue Jean Daniel Mukam Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise Results in Applied Mathematics Semilinear parabolic partial differential equations Finite element method Weak convergence Additive noise |
| title | Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise |
| title_full | Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise |
| title_fullStr | Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise |
| title_full_unstemmed | Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise |
| title_short | Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise |
| title_sort | weak convergence of the finite element method for semilinear parabolic spdes driven by additive noise |
| topic | Semilinear parabolic partial differential equations Finite element method Weak convergence Additive noise |
| url | http://www.sciencedirect.com/science/article/pii/S2590037422000747 |
| work_keys_str_mv | AT antoinetambue weakconvergenceofthefiniteelementmethodforsemilinearparabolicspdesdrivenbyadditivenoise AT jeandanielmukam weakconvergenceofthefiniteelementmethodforsemilinearparabolicspdesdrivenbyadditivenoise |