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A Two-Level Sparse Grid Collocation Method for Semilinear Stochastic Elliptic Equation
| Content Provider | Scilit |
|---|---|
| Author | Chen, Luoping Chen, Yanping Liu, Xiong |
| Copyright Year | 2017 |
| Abstract | In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯 H ( D ) {\mathcal{T}_{H}(D)} with small scale of sparse collocation points η ( L , N ) {\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯 h ( D ) {\mathcal{T}_{h}(D)} using large scale of sparse collocation points η ( ℓ , N ) {\eta(\ell,N)} (where η ( L , N ) , η ( ℓ , N ) {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L , ℓ {L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h ≈ H 3 , η ( ℓ , N ) ≈ ( η ( L , N ) ) 3 {{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}} , the novel two-level discretization method achieves the same convergence accuracy in norm ∥ ⋅ ∥ ℒ ρ 2 ( Γ ) ⊗ ℒ 2 ( D ) {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ( ℒ ρ 2 ( Γ ) {\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ 2 {\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯 h ( D ) {\mathcal{T}_{h}(D)} and large scale collocation points η ( ℓ , N ) {\eta(\ell,N)} in random spaces. |
| Related Links | http://www.degruyter.com/downloadpdf/j/cmam.ahead-of-print/cmam-2017-0025/cmam-2017-0025.xml |
| Ending Page | 179 |
| Page Count | 15 |
| Starting Page | 165 |
| ISSN | 16094840 |
| e-ISSN | 16099389 |
| DOI | 10.1515/cmam-2017-0025 |
| Journal | Computational Methods in Applied Mathematics |
| Issue Number | 2 |
| Volume Number | 18 |
| Language | English |
| Publisher | Walter de Gruyter GmbH |
| Publisher Date | 2018-04-01 |
| Access Restriction | Open |
| Subject Keyword | Computational Methods in Applied Mathematics Mathematical Physics Stochastic Collocation Method Finite Element Method Semilinear Equation Convergence Analysis Journal: Computational Methods in Applied Mathematics, Issue- 7 |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics Numerical Analysis Computational Mathematics |
