An Adaptive Finite Element Method for the Laplace-Beltrami Operator on Implicitly Defined Surfaces

We present an adaptive finite element method for approximating solutions to the Laplace-Beltrami equation on surfaces in $\mathbb{R}^3$ which may be implicitly represented as level sets of smooth functions. Residual-type a posteriori error bounds which show that the error may be split into a “residual part” and a “geometric part” are established. In addition, implementation issues are discussed and several computational examples are given.

[1]  Willy Dörfler,et al.  An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation , 1998, Math. Comput..

[2]  R. Verfürth.  A posteriori error estimators for the Stokes equations , 1989 .

[3]  G. Dziuk,et al.  An algorithm for evolutionary surfaces , 1990 .

[4]  T. Apel,et al.  Clement-type interpolation on spherical domains—interpolation error estimates and application to a posteriori error estimation , 2005 .

[5]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[6]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[7]  G. Dziuk.  Finite Elements for the Beltrami operator on arbitrary surfaces , 1988 .

[8]  Michael J. Holst,et al.  Adaptive Numerical Treatment of Elliptic Systems on Manifolds , 2001, Adv. Comput. Math..

[9]  Andreas Veeser,et al.  A posteriori error estimators, gradient recovery by averaging, and superconvergence , 2006, Numerische Mathematik.

[10]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

[11]  Carsten Carstensen,et al.  Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis , 2004, Numerische Mathematik.

[12]  Martin Rumpf,et al.  A finite element method for surface restoration with smooth boundary conditions , 2004, Comput. Aided Geom. Des..

[13]  Willy Dörfler,et al.  An adaptive finite element method for a linear elliptic equation with variable coefficients , 2000 .

[14]  Ricardo H. Nochetto,et al.  A finite element method for surface diffusion: the parametric case , 2005 .

[15]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods , 2002, SIAM Rev..

[16]  Kunibert G. Siebert,et al.  Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA , 2005, Lecture Notes in Computational Science and Engineering.

[17]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs , 2005, SIAM J. Numer. Anal..