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

TL;DR

This paper introduces a reduced basis method for efficiently solving convection-diffusion equations with non-linear boundary conditions, enabling reuse of basis functions across similar problems and reducing computational complexity.

Contribution

A novel reduced basis approach that computes boundary Green's functions independently of non-linearities, applicable to drift-diffusion problems with non-linear boundary reactions.

Findings

Basis functions are reusable for similar problems.

Significant reduction in computational effort.

Effective for catalytic reaction modeling.

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.