An Adaptive Surface Finite Element Method Based on Volume Meshes

@article{Demlow2012AnAS,
  title={An Adaptive Surface Finite Element Method Based on Volume Meshes},
  author={Alan Demlow and Maxim A. Olshanskii},
  journal={SIAM J. Numer. Anal.},
  year={2012},
  volume={50},
  pages={1624-1647}
}
In this paper we define an adaptive version of a recently introduced finite element method for numerical treatment of elliptic PDEs defined on surfaces. The method makes use of a (standard) outer volume mesh to discretize an equation on a two-dimensional surface embedded in $\mathbb{R}^3$. Extension of the equation from the surface is avoided, but the number of degrees of freedom (d.o.f.) is optimal in the sense that it is comparable to methods in which the surface is meshed directly. In… 
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References

SHOWING 1-10 OF 27 REFERENCES
A finite element method for surface PDEs: matrix properties
TLDR
This paper addresses linear algebra aspects of this new finite element method for the discretization of elliptic partial differential equations on surfaces and proves that the (effective) spectral condition number of the diagonsally scaled mass matrix and the diagonally scaled stiffness matrix behaves like h.
A Finite Element Method for Elliptic Equations on Surfaces
TLDR
An analysis is given that shows that the method to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface has optimal order of convergence both in the H^1- and in the L^2-norm.
An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries
  • J. Greer
  • Mathematics, Computer Science
    J. Sci. Comput.
  • 2006
TLDR
The change remedies many of problems facing the original method, including a need to frequently extend data off of the surface, uncertain boundary conditions, and terribly degenerate parabolic PDEs.
Surface Finite Elements for Parabolic Equations
In this article we define a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces Γ in R. The key idea is based on the
Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces
  • A. Demlow
  • Mathematics, Computer Science
    SIAM J. Numer. Anal.
  • 2009
TLDR
Higher-order analogues to the piecewise linear surface finite element method studied in Dziuk's paper are defined and error estimates are proved in both pointwise and $L_2$-based norms.
Finite elements on evolving surfaces
In this article, we define a new evolving surface finite-element method for numerically approximating partial differential equations on hypersurfaces (t) in n+1 which evolve with time. The key idea
An h-narrow band finite-element method for elliptic equations on implicit surfaces
In this article we define a finite-element method for elliptic partial differential equations (PDEs) on curves or surfaces, which are given implicitly by some level set function. The method is
Finite element approximation of elliptic partial differential equations on implicit surfaces
The aim of this paper is to investigate finite element methods for the solution of elliptic partial differential equations on implicitly defined surfaces. The problem of solving such equations
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
Parallel Multilevel Tetrahedral Grid Refinement
TLDR
A new data distribution format is introduced that is very suitable for the parallel multilevel refinement algorithm and it is proved that the application of the parallel refinement algorithm to an input admissible hierarchical decomposition yields an admissible hierarchy decomposition.
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