[Paper Review] lymph: discontinuous poLYtopal methods for Multi-PHysics differential problems

Paola F. Antonietti, Stefano Bonetti+4 more|arXiv (Cornell University)|Jan 24, 2024

Advanced Numerical Methods in Computational Mathematics4 citations

TL;DR

This paper introduces lymph, an open-source MATLAB library for high-order discontinuous Galerkin (PolyDG) discretization on polytopal meshes to solve coupled multi-physics problems. It combines geometric flexibility, high-order accuracy, and robustness to heterogeneous material properties, demonstrating convergence and efficiency across Poisson, heat, and elastodynamics problems with complex interfaces and varying mesh resolution.

ABSTRACT

We present the library lymph for the finite element numerical discretization of coupled multi-physics problems. lymph is a Matlab library for the discretization of partial differential equations based on high-order discontinuous Galerkin methods on polytopal grids (PolyDG) for spatial discretization coupled with suitable finite-difference time marching schemes. The objective of the paper is to introduce the library by describing it in terms of installation, input/output data, and code structure, highlighting - when necessary - key implementation aspects related to the method. A user guide, proceeding step-by-step in the implementation and solution of a Poisson problem, is also provided. In the last part of the paper, we show the results obtained for several differential problems, namely the Poisson problem, the heat equation, the elastodynamics system, and a multiphysics problem coupling poroelasticity and acoustic equations. Through these examples, we show the convergence properties and highlight some of the main features of the proposed method, i.e. geometric flexibility, high-order accuracy, and robustness with respect to heterogeneous physical parameters.

Motivation & Objective

• To develop a flexible, user-friendly MATLAB library for solving coupled multi-physics problems using high-order discontinuous Galerkin methods on polytopal grids.
• To enable accurate and robust numerical simulations in complex geometries with heterogeneous material properties and non-conforming interfaces.
• To provide a general-purpose, extensible framework that supports arbitrary-order spatial discretization and time integration via finite differences.
• To demonstrate the method's convergence, scalability, and robustness across benchmark problems including Poisson, heat, and elastodynamics systems.
• To offer a complete software stack with installation, input/output handling, and a step-by-step user guide for reproducible simulations.

Proposed method

• Spatial discretization is performed using high-order discontinuous Galerkin methods on polytopal (polygonal/polyhedral) meshes, enabling geometric flexibility and local refinement.
• The method employs hierarchical polynomial bases and local solvers that are independent of the number of element faces, supporting agglomeration and h-refinement.
• Time integration is handled via Crank-Nicolson for first-order systems and Newmark-β for second-order systems, ensuring stability and accuracy in time.
• The library supports arbitrary-order accuracy and is dimension-independent, facilitating seamless transition from 2D to 3D problems.
• Implementation is modular and extensible, allowing users to couple existing physics or implement new ones with minimal code changes.
• The software includes a complete user guide with a step-by-step solution of a Poisson problem, ensuring accessibility for new users.

Experimental results

Research questions

• RQ1 How can high-order discontinuous Galerkin methods on polytopal meshes be effectively implemented in a general-purpose, user-friendly MATLAB library for multi-physics problems?
• RQ2 What is the convergence behavior and computational efficiency of the PolyDG method when applied to coupled problems with heterogeneous material properties?
• RQ3 How does the library handle complex geometries, non-conforming interfaces, and varying mesh resolution in multi-physics simulations?
• RQ4 Can the library achieve robust performance and high-order accuracy across diverse PDEs, including elliptic, parabolic, and hyperbolic systems?
• RQ5 What are the practical computational costs and scalability characteristics of the method in realistic 3D multi-material settings?

Key findings

• The lymph library achieves optimal convergence rates for the Poisson, heat, and elastodynamics problems, confirming the theoretical accuracy of the PolyDG method.
• For the elastodynamics test case with 409,080 degrees of freedom, the assembly of system matrices took 2056 seconds, and a single time step required approximately 300 seconds.
• The method successfully captures wave propagation with strong perturbations and surface waves due to material discontinuities, demonstrating robustness in heterogeneous media.
• The use of polytopal meshes with agglomerated elements allows for local refinement and complex interface representation without compromising solver efficiency.
• The library exhibits strong geometric flexibility, enabling accurate modeling of complex domains with varying mesh sizes (from h ≈ 160 m to h ≈ 1500 m) across different materials.
• The implementation supports arbitrary polynomial degrees (e.g., ℓ = 5) and maintains stability and accuracy across all test problems, including those with high material contrast.