Robust globally divergence-free weak Galerkin methods for unsteady incompressible convective Brinkman-Forchheimer equations

TL;DR

This paper introduces and analyzes a class of weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations, achieving divergence-free velocity approximation and optimal error estimates.

Contribution

The paper develops a novel weak Galerkin approach with divergence-free velocity approximation and provides rigorous error analysis and an efficient iterative solver.

Findings

Methods achieve globally divergence-free velocity fields.

Optimal error estimates are established in energy and L2 norms.

Numerical experiments confirm theoretical convergence and accuracy.

Abstract

This paper develops and analyzes a class of semi-discrete and fully discrete weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations. For the spatial discretization, the methods adopt the piecewise polynomials of degrees $m\ (m\geq1)$ and $m-1$ respectively to approximate the velocity and pressure inside the elements, and piecewise polynomials of degree $m$ to approximate their numerical traces on the interfaces of elements. In the fully discrete method, the backward Euler difference scheme is used to approximate the time derivative. The methods are shown to yield globally divergence-free velocity approximation. Optimal a priori error estimates in the energy norm and $L^2$ norm are established. A convergent linearized iterative algorithm is designed for solving the fully discrete system. Numerical experiments are provided to verify the theoretical results.