Porous Convection in the Discrete Exterior Calculus with Geometric Multigrid

2025-08-17

AI Summary

This work addresses the efficient numerical solution of convection in porous media. We propose the first geometric multigrid method within the discrete exterior calculus (DEC) framework. The method constructs structure-preserving discrete formulations on simplicial complexes, ensures topologically consistent discretization via CombinatorialSpaces.jl, and couples multiphysics equations using Decapodes.jl. Crucially, we introduce a novel geometric mapping between refined simplicial complexes, enabling coordinated transfer of DEC operators across multiresolution grids—supporting both standalone solving and preconditioning. Our Julia-based multigrid solver demonstrates optimal-order convergence and substantial speedup for both Poisson equations and porous convection problems. These results validate the method’s effectiveness and scalability for structure-preserving numerical simulation.

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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.

Problem

• Formulate porous convection equations using discrete exterior calculus
• Implement geometric multigrid solver for simplicial complexes
• Demonstrate solver performance for Poisson and porous convection problems

Innovation

• Discrete exterior calculus for porous convection
• Geometric multigrid solver implementation
• Julia-based eDSL for multiphysics problems