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An optimally stable approximation of reactive transport using discrete test and infinite trial spaces

Lukas Renelt, Christian Engwer, Mario Ohlberger

TL;DR

The paper tackles stability challenges in stationary reactive transport by formulating an optimally stable ultraweak Petrov–Galerkin method that discretizes only the test space while the trial space remains infinite, yielding a normal equation defined on the test space and enabling post-processed functional reconstructions. The framework relies on an adjoint operator $A^*$ and isometries between the trial and test spaces, achieving conditioning with $\kappa=1$ in the continuous setting and a discrete normal equation that remains well-posed under discretization. A discrete test-space only strategy is developed, avoiding explicit construction of a trial-space discretization and permitting functional evaluations (e.g., point values) of the solution directly from $w^\delta$, with numerical experiments in catalytic-filter models illustrating convergence rates and practical performance. The results indicate a robust, efficiently solvable approach with potential for reduction to reduced-order models, while highlighting ongoing challenges in preconditioning and solving the resulting linear systems for complex velocity fields and non-constant data. Overall, the work provides a theoretically grounded, computation-friendly pathway for stable reactive-transport simulations where functionals of the solution are of primary interest.

Abstract

In this contribution we propose an optimally stable ultraweak Petrov-Galerkin variational formulation and subsequent discretization for stationary reactive transport problems. The discretization is exclusively based on the choice of discrete approximate test spaces, while the trial space is a priori infinite dimensional. The solution in the trial space or even only functional evaluations of the solution are obtained in a post-processing step. We detail the theoretical framework and demonstrate its usage in a numerical experiment that is motivated from modeling of catalytic filters.

An optimally stable approximation of reactive transport using discrete test and infinite trial spaces

TL;DR

The paper tackles stability challenges in stationary reactive transport by formulating an optimally stable ultraweak Petrov–Galerkin method that discretizes only the test space while the trial space remains infinite, yielding a normal equation defined on the test space and enabling post-processed functional reconstructions. The framework relies on an adjoint operator and isometries between the trial and test spaces, achieving conditioning with in the continuous setting and a discrete normal equation that remains well-posed under discretization. A discrete test-space only strategy is developed, avoiding explicit construction of a trial-space discretization and permitting functional evaluations (e.g., point values) of the solution directly from , with numerical experiments in catalytic-filter models illustrating convergence rates and practical performance. The results indicate a robust, efficiently solvable approach with potential for reduction to reduced-order models, while highlighting ongoing challenges in preconditioning and solving the resulting linear systems for complex velocity fields and non-constant data. Overall, the work provides a theoretically grounded, computation-friendly pathway for stable reactive-transport simulations where functionals of the solution are of primary interest.

Abstract

In this contribution we propose an optimally stable ultraweak Petrov-Galerkin variational formulation and subsequent discretization for stationary reactive transport problems. The discretization is exclusively based on the choice of discrete approximate test spaces, while the trial space is a priori infinite dimensional. The solution in the trial space or even only functional evaluations of the solution are obtained in a post-processing step. We detail the theoretical framework and demonstrate its usage in a numerical experiment that is motivated from modeling of catalytic filters.
Paper Structure (10 sections, 6 theorems, 24 equations, 3 figures, 1 table)

This paper contains 10 sections, 6 theorems, 24 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Ultraweak Petrov-Galerkin variational formulation and related normal equation
  3. Choice of appropriate function spaces
  4. Optimally stable ultraweak formulation and normal equation
  5. A test-space only discretization
  6. Discrete normal equation and functional reconstruction
  7. Conditioning of the system matrix and solving the linear system
  8. Numerical experiments
  9. Conclusion
  10. Acknowledgements

Key Result

proposition thmcounterproposition

There exists a linear and continuous trace operator as well as a linear and continuous projection operator fulfilling $\gamma(u) = u\raisebox{-.5ex}{$|$}_{\Gamma_+}$ and $\mathrm{pr}_{L^2}(u) = u$ for all $u\in U \subset \mathcal{X}$.

Figures (3)

  • Figure 1: Problem setup and Darcy velocity field $\hbox{\boldmath$\mathbf{b}$}$
  • Figure 2: Solutions obtained by different discretization methods. All of them are based on a structured grid with $h^{-1}=40$.
  • Figure 3: Convergence under $h$-refinement

Theorems & Definitions (10)

  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • corollary thmcountercorollary
  • proposition thmcounterproposition
  • definition thmcounterdefinition: Ultraweak variational formulation of reactive transport
  • proposition thmcounterproposition
  • proof
  • corollary thmcountercorollary: Optimal stability
  • definition thmcounterdefinition: Normal equation of the ultraweak formulation
  • remark thmcounterremark