Volume 25, Issue 4
Surface Finite Elements for Parabolic Equations
DOI:

J. Comp. Math., 25 (2007), pp. 385-407

Published online: 2007-08

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• Abstract

In this article we define a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\Gamma$ in $\mathbb R^{n+1}$. The key idea is based on the approximation of $\Gamma$ by a polyhedral surface $\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow.

• Keywords

Surface partial differential equations Surface finite element method Geodesic curvature Triangulated surface

65M60 65M30 65M12 65Z05 58J35 53A05 74S05 80M10 76M10.

@Article{JCM-25-385, author = {}, title = {Surface Finite Elements for Parabolic Equations}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {4}, pages = {385--407}, abstract = { In this article we define a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\Gamma$ in $\mathbb R^{n+1}$. The key idea is based on the approximation of $\Gamma$ by a polyhedral surface $\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8700.html} }
TY - JOUR T1 - Surface Finite Elements for Parabolic Equations JO - Journal of Computational Mathematics VL - 4 SP - 385 EP - 407 PY - 2007 DA - 2007/08 SN - 25 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/jcm/8700.html KW - Surface partial differential equations KW - Surface finite element method KW - Geodesic curvature KW - Triangulated surface AB - In this article we define a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\Gamma$ in $\mathbb R^{n+1}$. The key idea is based on the approximation of $\Gamma$ by a polyhedral surface $\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow.