Mesh-free mixed finite element approximation for nonlinear time-fractional biharmonic problem using weighted b-splines
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929281083375616 |
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| author | Mandaliya, Jitesh P. Kumar, Dileep Chaudhary, Sudhakar |
| author_facet | Mandaliya, Jitesh P. Kumar, Dileep Chaudhary, Sudhakar |
| contents | In this article, we propose a fully-discrete scheme for the numerical solution of a nonlinear time-fractional biharmonic problem. This problem is first converted into an equivalent system by introducing a new variable. Then spatial and temporal discretizations are done by the weighted $b$-spline method and $L2$-$1_σ$ approximation, respectively. The weighted $b$-spline method uses weighted $b$-splines on a tensor product grid as basis functions for the finite element space and by construction, it is a mesh-free method. This method combines the computational benefits of $b$-splines and standard mesh-based elements. We derive $α$-robust \emph{a priori} bound and convergence estimate in the $L^2(Ω)$ norm for the proposed scheme. Finally, we carry out few numerical experiments to support our theoretical findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_11196 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mesh-free mixed finite element approximation for nonlinear time-fractional biharmonic problem using weighted b-splines Mandaliya, Jitesh P. Kumar, Dileep Chaudhary, Sudhakar Numerical Analysis In this article, we propose a fully-discrete scheme for the numerical solution of a nonlinear time-fractional biharmonic problem. This problem is first converted into an equivalent system by introducing a new variable. Then spatial and temporal discretizations are done by the weighted $b$-spline method and $L2$-$1_σ$ approximation, respectively. The weighted $b$-spline method uses weighted $b$-splines on a tensor product grid as basis functions for the finite element space and by construction, it is a mesh-free method. This method combines the computational benefits of $b$-splines and standard mesh-based elements. We derive $α$-robust \emph{a priori} bound and convergence estimate in the $L^2(Ω)$ norm for the proposed scheme. Finally, we carry out few numerical experiments to support our theoretical findings. |
| title | Mesh-free mixed finite element approximation for nonlinear time-fractional biharmonic problem using weighted b-splines |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2403.11196 |