Porous Convection in the Discrete Exterior Calculus with Geometric Multigrid

TL;DR

This paper develops a numerical framework using discrete exterior calculus and geometric multigrid methods to efficiently solve porous convection equations on complex geometries, enhancing computational physics tools.

Contribution

It introduces a novel combination of DEC with multigrid solvers in Julia, enabling efficient simulation of porous convection problems on subdivided simplicial complexes.

Findings

Multigrid solver effectively accelerates convergence for Poisson and convection problems.

DEC preserves geometric properties in discretization, improving accuracy.

Open-source Julia implementation facilitates multiphysics simulations.

Abstract

The discrete exterior calculus (DEC) defines a family of discretized differential operators which preserve certain desirable properties from the exterior calculus. We formulate and solve the porous convection equations in the DEC via the Decapodes.jl embedded domain-specific language (eDSL) for multiphysics problems discretized via CombinatorialSpaces.jl. CombinatorialSpaces.jl is an open-source Julia library which implements the DEC over simplicial complexes, and now offers a geometric multigrid solver over maps between subdivided simplicial complexes. We demonstrate numerical results of multigrid solvers for the Poisson problem and porous convection problem, both as a standalone solver and as a preconditioner for open-source Julia iterative methods libraries.