Journal: Numerical Algorithms

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

doi: 10.1007/s11075-024-01919-4

Figure Lengend Snippet: A conforming bulk-surface mesh in 2D generated with the VEMcomp function generate_mesh2d. The bulk domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} Ω is approximated by a polygonal mesh \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _h$$\end{document} Ω h composed of square elements (green) and more general polygonal elements (orange) close to the surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} Γ . The surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} Γ is approximated by a piecewise straight line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _h$$\end{document} Γ h (blue)

Article Snippet: We have introduced VEMcomp, a user-friendly MATLAB library for (i) polytopal bulk-surface mesh generation in 2D and 3D, (ii) matrix assembly for the lowest-order FEM and VEM, (iii) solving bulk, surface, and bulk-surface, elliptic and parabolic, linear and semilinear PDEs or systems of PDEs and (iv) post-processing the numerical solution, in terms of plotting and error evaluation.

Techniques: Generated

Journal: Numerical Algorithms

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

doi: 10.1007/s11075-024-01919-4

Figure Lengend Snippet: A conforming bulk-surface mesh in 3D generated with the VEMcomp function generate_mesh3d. The bulk domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} Ω is approximated by a polyhedral mesh \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _h$$\end{document} Ω h composed of cubic elements (green) and more general polyhedral elements (orange) close to the surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} Γ . The surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} Γ is approximated by a triangulated surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _h$$\end{document} Γ h (blue) taken as the boundary of the bulk mesh \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _h$$\end{document} Ω h

Article Snippet: We have introduced VEMcomp, a user-friendly MATLAB library for (i) polytopal bulk-surface mesh generation in 2D and 3D, (ii) matrix assembly for the lowest-order FEM and VEM, (iii) solving bulk, surface, and bulk-surface, elliptic and parabolic, linear and semilinear PDEs or systems of PDEs and (iv) post-processing the numerical solution, in terms of plotting and error evaluation.

Techniques: Generated

Journal: Numerical Algorithms

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

doi: 10.1007/s11075-024-01919-4

Figure Lengend Snippet: Numerical solution of the elliptic bulk problem obtained in VEMcomp on a mesh with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=1336$$\end{document} N = 1336 nodes and meshsize \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=0.0725$$\end{document} h = 0.0725

Article Snippet: We have introduced VEMcomp, a user-friendly MATLAB library for (i) polytopal bulk-surface mesh generation in 2D and 3D, (ii) matrix assembly for the lowest-order FEM and VEM, (iii) solving bulk, surface, and bulk-surface, elliptic and parabolic, linear and semilinear PDEs or systems of PDEs and (iv) post-processing the numerical solution, in terms of plotting and error evaluation.

Techniques:

Journal: Numerical Algorithms

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

doi: 10.1007/s11075-024-01919-4

Figure Lengend Snippet: Numerical solution of the elliptic bulk problem obtained in VEMcomp on a mesh with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=16600$$\end{document} N = 16600 nodes and meshsize \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=0.1195$$\end{document} h = 0.1195 . The colormap is different than in the other experiments, in order to better highlight the structure of the mesh

Article Snippet: We have introduced VEMcomp, a user-friendly MATLAB library for (i) polytopal bulk-surface mesh generation in 2D and 3D, (ii) matrix assembly for the lowest-order FEM and VEM, (iii) solving bulk, surface, and bulk-surface, elliptic and parabolic, linear and semilinear PDEs or systems of PDEs and (iv) post-processing the numerical solution, in terms of plotting and error evaluation.

Techniques:

Journal: Numerical Algorithms

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

doi: 10.1007/s11075-024-01919-4

Figure Lengend Snippet: Numerical solution of the parabolic surface PDE obtained in VEMcomp on a mesh with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=3888$$\end{document} N = 3888 nodes and meshsize \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=0.1195$$\end{document} h = 0.1195

Article Snippet: We have introduced VEMcomp, a user-friendly MATLAB library for (i) polytopal bulk-surface mesh generation in 2D and 3D, (ii) matrix assembly for the lowest-order FEM and VEM, (iii) solving bulk, surface, and bulk-surface, elliptic and parabolic, linear and semilinear PDEs or systems of PDEs and (iv) post-processing the numerical solution, in terms of plotting and error evaluation.

Techniques:

Journal: Numerical Algorithms

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

doi: 10.1007/s11075-024-01919-4

Figure Lengend Snippet: Numerical solution at the final time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=10$$\end{document} T = 10 of the S-RDS on the ellipsoid \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} Γ defined in , solved in VEMcomp on a mesh with 3990 nodes and meshsize \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h \le 0.1$$\end{document} h ≤ 0.1 , whose exact value is not provided by DistMesh. Top row: component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1$$\end{document} v 1 (left) and component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_2$$\end{document} v 2 (right). Bottom row: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} L 2 norm of time derivative of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1$$\end{document} v 1 over time (left) and spatial average of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1$$\end{document} v 1 over time (right)

Article Snippet: We have introduced VEMcomp, a user-friendly MATLAB library for (i) polytopal bulk-surface mesh generation in 2D and 3D, (ii) matrix assembly for the lowest-order FEM and VEM, (iii) solving bulk, surface, and bulk-surface, elliptic and parabolic, linear and semilinear PDEs or systems of PDEs and (iv) post-processing the numerical solution, in terms of plotting and error evaluation.

Techniques:

Journal: Numerical Algorithms

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

doi: 10.1007/s11075-024-01919-4

Figure Lengend Snippet: Elliptic bulk-surface problem on the unit sphere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} Ω in 3D, solved in VEMcomp on a mesh with 16600 nodes and meshsize 0.1195. Left: bulk component u . Right: surface component v

Article Snippet: We have introduced VEMcomp, a user-friendly MATLAB library for (i) polytopal bulk-surface mesh generation in 2D and 3D, (ii) matrix assembly for the lowest-order FEM and VEM, (iii) solving bulk, surface, and bulk-surface, elliptic and parabolic, linear and semilinear PDEs or systems of PDEs and (iv) post-processing the numerical solution, in terms of plotting and error evaluation.

Techniques:

Journal: Numerical Algorithms

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

doi: 10.1007/s11075-024-01919-4

Figure Lengend Snippet: Numerical solution at the final time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=5$$\end{document} T = 5 of the BS-RDS on the torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} Ω defined in , solved in VEMcomp on a mesh with 8144 nodes and meshsize 0.1074. Top row: bulk components \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_1,u_2)$$\end{document} ( u 1 , u 2 ) . Bottom row: surface components \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v_1,v_2)$$\end{document} ( v 1 , v 2 )

Article Snippet: We have introduced VEMcomp, a user-friendly MATLAB library for (i) polytopal bulk-surface mesh generation in 2D and 3D, (ii) matrix assembly for the lowest-order FEM and VEM, (iii) solving bulk, surface, and bulk-surface, elliptic and parabolic, linear and semilinear PDEs or systems of PDEs and (iv) post-processing the numerical solution, in terms of plotting and error evaluation.

Techniques: