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A bound-preserving and conservative enriched Galerkin method for elliptic problems

Gabriel R. Barrenechea, Philip L. Lederer, Andreas Rupp

TL;DR

Bound-preserving discretization for elliptic reaction-diffusion problems is developed by augmenting the enriched Galerkin method with over-penalized jump terms and a nonlinear projection that splits the solution into linear and piecewise-constant components. A nonlinear fixed-point/decoupling strategy yields well-conditioned subproblems while guaranteeing that the discrete solution remains within the physical bounds and preserves local mass conservation. The authors prove existence of discrete, bound-preserving solutions and derive near-optimal error estimates, showing robust convergence when the penalty parameter satisfies $eta\ge 4$; numerical experiments demonstrate strong bound preservation, mass conservation, and efficient iterations across varied parameters. The approach offers a practical and theoretically sound framework for physically consistent simulations with conservative properties, and opens avenues for adaptive parameter choices and extensions to nonlinear or hyperbolic settings.

Abstract

We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial over-penalization of the discrete solution's jumps to obtain optimal convergence. To avoid the ill-conditioning issues that arise in over-penalized schemes, we introduce an involved splitting approach that separates the system of equations for the discontinuous solution part from the system of equations for the continuous solution part, yielding well-behaved subproblems. We prove the existence of discrete solutions and optimal error estimates, which are validated numerically.

A bound-preserving and conservative enriched Galerkin method for elliptic problems

TL;DR

Bound-preserving discretization for elliptic reaction-diffusion problems is developed by augmenting the enriched Galerkin method with over-penalized jump terms and a nonlinear projection that splits the solution into linear and piecewise-constant components. A nonlinear fixed-point/decoupling strategy yields well-conditioned subproblems while guaranteeing that the discrete solution remains within the physical bounds and preserves local mass conservation. The authors prove existence of discrete, bound-preserving solutions and derive near-optimal error estimates, showing robust convergence when the penalty parameter satisfies ; numerical experiments demonstrate strong bound preservation, mass conservation, and efficient iterations across varied parameters. The approach offers a practical and theoretically sound framework for physically consistent simulations with conservative properties, and opens avenues for adaptive parameter choices and extensions to nonlinear or hyperbolic settings.

Abstract

We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial over-penalization of the discrete solution's jumps to obtain optimal convergence. To avoid the ill-conditioning issues that arise in over-penalized schemes, we introduce an involved splitting approach that separates the system of equations for the discontinuous solution part from the system of equations for the continuous solution part, yielding well-behaved subproblems. We prove the existence of discrete solutions and optimal error estimates, which are validated numerically.

Paper Structure

This paper contains 20 sections, 17 theorems, 67 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model problem and baseline discretization
  3. Positivity-preserving finite elements
  4. The nonlinear finite element method
  5. Existence of a discrete solution
  6. Preliminaries and auxiliary results
  7. Well-posedness of the iteration defined in \ref{['EQ:fixed_point']}
  8. Proof of Theorem \ref{['TH:existence']}
  9. Worst-case criterion for positivity preservation
  10. Convergence order estimates
  11. Numerical examples
  12. An iterative algorithm
  13. The condition number of the linearized problems
  14. Numerical investigation:
  15. Smooth solution
  16. ...and 5 more sections

Key Result

Lemma 3.1

Let $v_h \in V_h$. If $a - \underline v_{hi} \le b - \overline v_{hi}$ for all node indices $i = 1, \dots, N$, then $v^+_h(\boldsymbol{x}) \in \mathcal{G}$ for all $\boldsymbol{x}\in\Omega$.

Figures (4)

  • Figure 1: Illustration of $P_h^{v_h}$ on three elements $T_1, T_2, T_3$. Left: $v_h$ (cyan) and $v^1_h$ (orange) and $v_h^{0}$ (purple). Right: $v_h^+$ (cyan) and $v_h^{1+}$ (orange) with $v_h^-$ (purple). In both cases, $a=0$ and $b=1$.
  • Figure 1: Number of iterations for the example of Section \ref{['sec:smooth_solution']} for different penalty factors $\gamma$ and with $tol_n = 10^{-9}$ and $tol_m = 10^{-12}$.
  • Figure 2: Number of iterations for the example of Section \ref{['sec:smooth_solution']} for different factors $\beta$ and with $tol_n = 10^{-9}$ and $tol_m = 10^{-12}$.
  • Figure 3: Solutions of the interior layer example of Section \ref{['sec:interior_layer']} for the standard EG method (left) and the proposed method with $\gamma = 10$ (right). For better visualization, the solution on the partial domain $[0,1] \times [0.5,1]$ is shown here.

Theorems & Definitions (33)

  • Remark 2.1: Properties of the EG method
  • Lemma 3.1
  • Remark 3.2: Bound preservation and mass conservation
  • Theorem 4.1
  • Lemma 4.2: Broken Poincaré inequality, Brenner03
  • Lemma 4.3
  • Proof 1
  • Lemma 4.4
  • Proof 2
  • Lemma 4.5
  • ...and 23 more