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_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 a priori bound and convergence estimate in the L²(Ω) norm for the proposed scheme. Finally, we carry out few numerical experiments to support our theoretical findings.