Space-Time Continuous and Time Discontinuous Galerkin Schemes Based on Isogeometric Analysis for Nonlinear Time-Fractional Partial Differential Equations (Journal of Computational Mathematics, 2024)
This paper presents space-time continuous and time discontinuous Galerkin schemesfor solving nonlinear time-fractional partial differential equations based on B-splines intime and non-uniform rational B-splines (NURBS) in space within the framework of Isogeometric Analysis. The first approach uses the space-time continuous Petrov-Galerkintechnique for a class of nonlinear time-fractional Sobolev-type equations and the optimalerror estimates are obtained through a concise equivalence analysis. The second approachemploys a generalizable time discontinuous Galerkin scheme for the time-fractional Allen-Cahn equation. It first transforms the equation into a time integral equation and then usesthe discontinuous Galerkin method in time and the NURBS discretization in space. Theoptimal error estimates are provided for the approach. The convergence analysis undertime graded meshes is also carried out, taking into account the initial singularity of thesolution for two models. Finally, numerical examples are presented to demonstrate theeffectiveness of the proposed methods.