Automatic Variationally Stable Analysis for Finite Element Computations: Transient Convection-Diffusion Problems

TL;DR

This paper introduces an unconditionally stable finite element method for transient convection-diffusion problems, enabling flexible discretization in space and time with proven convergence and error estimation capabilities.

Contribution

The paper develops an automatic variationally stable finite element method that guarantees stability regardless of the differential operator, applicable to space-time and method of lines discretizations.

Findings

Unconditional stability of the AVS-FE method for convection-diffusion problems.

Optimal convergence demonstrated through numerical studies.

Effective a posteriori error estimates for adaptive refinement.

Abstract

We establish stable finite element (FE) approximations of convection-diffusion initial boundary value problems using the automatic variationally stable finite element (AVS-FE) method. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered singular perturbation in both space and time. The unconditional stability of the AVS-FE method, regardless of the underlying differential operator, allows us significant flexibility in the construction of FE approximations. We take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-alpha method. In the generalized-alpha method, we discretize the temporal domain into finite sized time-steps and adopt the generalized-alpha method as time integrator. Then, we derive a corresponding norm for the obtained operator to guarantee the temporal stability of the method. We present numerical verifications for both approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin method by Demkowicz and Gopalakrishnan, the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serve as error indicators in both space and time for which we present multiple numerical verifications adaptive strategies.