lymph: discontinuous poLYtopal methods for Multi-PHysics differential problems

TL;DR

The library lymph is a MATLAB library for the discretization of partial differential equations based on high-order discontinuous Galerkin methods on polytopal grids for spatial discretization coupled with suitable finite-difference time marching schemes and the results obtained are shown.

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.

Figures (10)

• Figure 1: lymph code structure and logo (top left).
• Figure 2: Test case of Section \ref{['sec:poisson']}. Left: computed PolyDG solution $u_h$ using a polygonal mesh with $N_{el} = 30$ elements and a polynomial degree $\ell = 3$. Center: analytical solution $u_{ex}$. Right: the difference between numerical and analytical solutions.
• Figure 3: Test case of Section \ref{['sec:poisson']}. Left: computed errors $\|u-u_h\|_{dG}$ and $\|u-u_h\|_{L^2(\Omega)}$ as a function of the mesh size $h$ by fixing the polynomial degree $\ell=4$. Right: computed errors $\|u-u_h\|_{dG}$ and $\|u-u_h\|_{L^2(\Omega)}$ as a function of the polynomial degree $\ell=4$ by fixing the number of mesh element $N_{el}=100$.
• Figure 4: Test case of Section \ref{['sec:poisson']}. Triangular mesh of brain section (left), agglomerated polygonal mesh (center), and computed PolyDG solution on the polygonal grid with $\ell=1$ (right).
• Figure 5: Test case of Section \ref{['sec:poisson']}. Computed approximation errors in the $L^2$-norm on the triangular (red) and agglomerated (blue) meshes in Figure \ref{['fig:sol_lap_comp']}. Computed errors versus the total number of degrees of freedom (left), and versus the computational time (right).

Theorems & Definitions (1)

• Definition 2.1: Polytopic-regular mesh