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VEMcomp: a Virtual Elements MATLAB package for bulk-surface PDEs in 2D and 3D.

Massimo Frittelli1, Anotida Madzvamuse2,3,4,5, Ivonne Sgura1

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Numerical Algorithms
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Summary
This summary is machine-generated.

This study introduces VEMcomp, a Virtual Element Method (VEM) MATLAB solver for Partial Differential Equations (PDEs). VEMcomp efficiently handles complex geometries and various PDE types, offering a versatile tool for scientific computing.

Keywords:
Bulk-surface PDEsBulk-surface finite element methodBulk-surface virtual element methodIMEX Euler MethodMATLABMesh generationVirtual element method

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Scientific Computing

Background:

  • Partial Differential Equations (PDEs) are fundamental in modeling phenomena across material sciences, engineering, and biology.
  • Existing numerical methods often face challenges with complex geometries and diverse PDE formulations.
  • A need exists for flexible and efficient solvers applicable to a wide range of scientific problems.

Purpose of the Study:

  • To present VEMcomp, a Virtual Element MATLAB solver for elliptic and parabolic, linear and semilinear PDEs in 2D and 3D.
  • To provide a unified library for generating meshes, computing matrices, and solving PDE problems.
  • To offer a user-friendly interface for applying the Virtual Element Method (VEM) and Finite Element Method (FEM) to various PDE models.

Main Methods:

  • Implementation of the Virtual Element Method (VEM) of lowest polynomial order (k=1) on general polygonal and polyhedral meshes.
  • Integration of mesh generation capabilities for polygonal and polyhedral domains, including compatibility with DistMesh for surface PDEs.
  • Utilization of IMEX Euler for time-stepping in transient problems, coupled with VEM/FEM spatial discretization.

Main Results:

  • VEMcomp successfully generates optimized meshes and computes necessary matrices (mass and stiffness) for VEM and FEM.
  • The solver handles linear and nonlinear models, including time-dependent bulk, surface, and bulk-surface PDEs.
  • Optional post-processing functions for visualization and error estimation are provided.

Conclusions:

  • VEMcomp offers a powerful and flexible tool for solving a broad spectrum of PDEs in science and engineering.
  • The library's ability to handle complex geometries and various PDE types makes it valuable for researchers.
  • VEMcomp facilitates the application of VEM and FEM, enhancing numerical solutions for challenging problems.