Reduced Basis Approach for Convection-Diffusion Equations with Non-Linear Boundary Reaction Conditions

TL;DR

The paper addresses efficiently solving drift-diffusion problems with nonlinear boundary reactions, common in heterogeneous catalysis, by introducing a reduced-basis method that constructs discrete Green's-function–like bases on the boundary. The solution is split as $Y_h = Y_0 + Y_{ ext{nl}} = Y_0 + \sum_{K \in \mathcal{K}_{\text{nl}}} \alpha_K Y_K$, enabling an offline phase to build $Y_K$ and an online phase that solves a small boundary nonlinear system. The algebraic formulation leverages a common operator $A$ with linear solves for $x_0$ and $x_K$, while the nonlinear dependence is confined to a reduced system evaluated at boundary collocation points; the approach is demonstrated with a mass-conservative Voronoi-FVM discretization and a catalytic CO oxidation example, achieving machine-precision agreement with the full solution and roughly 100x online speedup. This framework is particularly suited for parameterized nonlinearity, inverse problems, and kinetic-model coupling, and it remains adaptable to velocity fields obtained from CFD and to basis compression techniques for further efficiency gains.

Abstract

This paper aims at an efficient strategy to solve drift-diffusion problems with non-linear boundary conditions as they appear, e.g., in heterogeneous catalysis. Since the non-linearity only involves the degrees of freedom along (a part of) the boundary, a reduced basis ansatz is suggested that computes discrete Green's-like functions for the present drift-diffusion operator such that the global non-linear problem reduces to a smaller non-linear problem for a boundary method. The computed basis functions are completely independent of the non-linearities. Thus, they can be reused for problems with the same differential operator and geometry. Corresponding scenarios might be inverse problems in heterogeneous catalysis but also modeling the effect of different catalysts in the same reaction chamber. The strategy is explained for a mass-conservative finite volume method and demonstrated on a simple numerical example for catalytic CO oxidation.

Figures (2)

• Figure 1: Computed mass fractions of $\mathrm{CO}$ and $\mathrm{CO}_2$ for $D = 10^{-2}$, $k = 10^{10}$ and $v_\mathrm{in} = 1$.
• Figure 2: Runtimes as a function of the refinement level (left) for $k = 10^{10}$ and number of Newton iterations as a function of $k$ (right), both for $D = 10^{-2}$ and $v_\mathrm{in} = 1$.

Theorems & Definitions (3)

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