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

Eirik Valseth, Pouria Behnoudfar, Clint Dawson, Albert Romkes

TL;DR

This work introduces the automatic variationally stable finite element (AVS-FE) method for transient convection-diffusion problems, addressing stability challenges arising from singular perturbations in space and time. It develops two discretization paths: a space-time AVS-FE approach and a method-of-lines strategy that uses AVS-FE in space together with the generalized-$\alpha$ time integrator, both yielding unconditional temporal stability. A key contribution is the construction of an energy norm and a residual-based error representation via the Riesz map, providing built-in a posteriori error estimates and indicators for space-time adaptivity. The framework is demonstrated through convergence studies and shock problems, with adaptive refinement guiding accuracy in both space and time; a time-slice variant is proposed to mitigate computational cost while preserving accuracy. These results underscore AVS-FE’s robustness and practical potential for solving challenging transient PDEs with rigorous error control.

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.

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

TL;DR

This work introduces the automatic variationally stable finite element (AVS-FE) method for transient convection-diffusion problems, addressing stability challenges arising from singular perturbations in space and time. It develops two discretization paths: a space-time AVS-FE approach and a method-of-lines strategy that uses AVS-FE in space together with the generalized- time integrator, both yielding unconditional temporal stability. A key contribution is the construction of an energy norm and a residual-based error representation via the Riesz map, providing built-in a posteriori error estimates and indicators for space-time adaptivity. The framework is demonstrated through convergence studies and shock problems, with adaptive refinement guiding accuracy in both space and time; a time-slice variant is proposed to mitigate computational cost while preserving accuracy. These results underscore AVS-FE’s robustness and practical potential for solving challenging transient PDEs with rigorous error control.

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.

Paper Structure

This paper contains 18 sections, 3 theorems, 45 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The AVS-FE Methodology
  3. Model Problem and Notation
  4. Weak Formulation and FE Discretization
  5. Saddle Point Problem
  6. Time Discretization
  7. Method of Lines
  8. Generalized-$\alpha$ and the AVS-FE Method
  9. Retrieving initial data
  10. Space-Time FE Approach
  11. Time Slice Approach
  12. Numerical Verifications
  13. Optimal Test Function Resolution
  14. Convergence Studies
  15. $H-div$ Conforming Basis Functions
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $(u, \mathbf{q} ) \in U(\Omega)$ be the solution of the AVS-FE weak form eq:weak_form and $(u^h, \mathbf{q} ^h) \in U^h(\Omega)$ its corresponding AVS-FE approximation through eq:discrete_form. Then, the the energy norm of $(u-u^h, \mathbf{q} - \mathbf{q}

Figures (8)

  • Figure 1: Partition of space-time domain into slices.
  • Figure 2: Convergence histories for the space-time convergence study.
  • Figure 3: Convergence study of the solution obtained using the generalized-$\alpha$ method for time discretization at final time $T=0.5\,s$.
  • Figure 4: Convergence study of the solution obtained using the generalized-$\alpha$ approach $T=1.0$ using fully conforming FE basis functions.
  • Figure 5: Convergence at the final time $T=0.5$ for increasingly fine uniform meshes.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Proposition 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • Proposition 3.1