DPG method with optimal test functions for a fractional advection diffusion equation

TL;DR

This paper introduces a DPG method with optimal test functions for a fractional advection diffusion equation, demonstrating its well-posedness, convergence, and effectiveness through numerical experiments.

Contribution

It develops an ultra-weak variational formulation and a DPG approximation with optimal test functions for fractional PDEs, ensuring quasi-optimal convergence.

Findings

Numerical experiments confirm convergence on uniform meshes.

Adaptive mesh refinement improves solution accuracy.

The method is well-posed and theoretically sound.

Abstract

We develop an ultra-weak variational formulation of a fractional advection diffusion problem in one space dimension and prove its well-posedness. Based on this formulation, we define a DPG approximation with optimal test functions and show its quasi-optimal convergence. Numerical experiments confirm expected convergence properties, for uniform and adaptively refined meshes.