Mesh-free mixed finite element approximation for nonlinear time-fractional biharmonic problem using weighted b-splines

Abstract

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.

Figures (4)

• Figure 1: Uniform $b$-spline
• Figure 2: $B$-spline on bounded domain
• Figure 3: Plot of the weight function (left) and numerical solution with its domain (right) for Example 1.
• Figure 4: Plot of the weight function (left) and numerical solution with its domain (right) for Example 2.

Theorems & Definitions (9)

• Definition 3.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Theorem 4.1
• Theorem 4.2
• Lemma 4.4
• Lemma 4.5
• Theorem 4.3