Mesh-free mixed finite element approximation for nonlinear time-fractional biharmonic problem using weighted b-splines
<|reference_start|>Mesh-free mixed finite element approximation for nonlinear time-fractional biharmonic problem using weighted b-splines: 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 $L2$-$1_\sigma$ 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 $\alpha$-robust \emph{a priori} bound and convergence estimate in the $L^2(\Omega)$ norm for the proposed scheme. Finally, we carry out few numerical experiments to support our theoretical findings.<|reference_end|>
@article{mandaliya2024mesh-free,
    title={Mesh-free mixed finite element approximation for nonlinear
  time-fractional biharmonic problem using weighted b-splines},
    author={Jitesh P. Mandaliya, Dileep Kumar, Sudhakar Chaudhary},
    journal={arXiv preprint arXiv:2403.11196},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.11196},
    primaryClass={math.NA cs.NA}
}