Variable Diffusion and Diffusion–Advection–Reaction

Abstract

In this chapter we extend the HHO method to the scalar diffusion–advection–reaction problem: Find $u:\Omega \to \mathbb {R}$ such that

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle \boldsymbol{\nabla}{\cdot}(-{\mathsf{K}}\boldsymbol{\nabla} u + \boldsymbol{\beta} u) + \mu u = f \qquad \text{in }\Omega{,} \\ &\displaystyle &\displaystyle \qquad \qquad \qquad u = 0 \qquad \text{on }\partial\Omega\text{,} \end{array} \end{aligned} $$

(3.1)

where ${\mathsf {K}}:\Omega \to \mathbb {R}_{\mathrm {sym}}^{d\times d}$ (with $\mathbb {R}_{\mathrm {sym}}^{d\times d}$ denoting the space of symmetric d × d matrices) is the spatially varying and possibly anisotropic diffusion coefficient, $\boldsymbol {\beta }:\Omega \to \mathbb {R}^d$ is the velocity, and $\mu :\Omega \to \mathbb {R}$ is the reaction coefficient.