Abstract:
We present a Virtual Element MATLAB solver for elliptic and parabolic, linear and
semilinear Partial Differential Equations (PDEs) in two and three space dimensions,
which is coined VEMcomp. Such PDEs are widely applicable to describing problems
in material sciences, engineering, cellular and developmental biology, among many
other applications. The library covers linear and nonlinear models posed on different
simple and complex geometries, involving time-dependent bulk, surface, and bulksurface
PDEs. The solver employs the Virtual Element Method (VEM) of lowest
polynomial order k = 1 on general polygonal and polyhedral meshes, including the
Finite Element Method (FEM) of order k = 1 as a special case when the considered
mesh is simplicial. VEMcomp has three main purposes. First, VEMcomp generates
polygonal and polyhedral meshes optimized for fast matrix assembly. Triangular and
tetrahedral meshes are encompassed as special cases. For surface PDEs, VEMcomp
is compatible with the well-known Matlab package DistMesh for mesh generation.
Second, given a mesh for the considered geometry, possibly generated with an external
package, VEMcomp computes all the matrices (mass and stiffness) required by the
VEM or FEM method. Third, for multiple classes of stationary and time-dependent
bulk, surface and bulk-surface PDEs, VEMcomp solves the considered PDE problem with the VEM or FEM in space and IMEX Euler in time, through a user-friendly
interface. As an optional post-processing, VEMcomp comes with its own functions for
plotting the numerical solutions and evaluating the error when possible. An extensive
set of examples illustrates the usage of the library.