# 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…

## Figures, Tables, and Topics from this paper

## 42 Citations

An adaptive octree finite element method for PDEs posed on surfaces

- Mathematics, Computer Science
- 2014

Analysis and numerical results suggest that combination of cartesian adaptive meshes and the unfitted (trace) finite elements provide simple, efficient, and reliable tool for numerical treatment of PDEs posed on surfaces.

A TRACE FINITE ELEMENT METHOD FOR PDES

- 2017

In this paper, we propose an approach for solving PDEs on evolving surfaces using a combination of the trace finite element method and a fast marching method. The numerical approach is based on the…

A Trace Finite Element Method for PDEs on Evolving Surfaces

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2017

An approach for solving PDEs on evolving surfaces using a combination of the trace finite element method and a fast marching method based on the Eulerian description of the surface problem and employs a time-independent background mesh that is not fitted to the surface.

L2 and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces

- Mathematics
- 2014

Surface Finite Element Methods (SFEM) are widely used to solve surface partial differential equations arising in applications including crystal growth, fluid mechanics and computer graphics. A…

A VOLUME MESH FINITE ELEMENT METHOD FOR PDES ON SURFACES

- Mathematics
- 2012

We treat a surface finite element method that is based on the trace of a standard finite element space on a tetrahedral triangulation of an outer domain that contains a stationary 2D surface. This…

A Finite Element Method for the Surface Stokes Problem

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2018

A Trace finite element method (TraceFEM) is developed and analyzed for a surface Stokes problem posed on a 2D surface embedded in a 3D domain and proves stability and optimal order discretization error bounds in the surface $H^1$ and $L^2$ norms.

A high‐order FEM with exact geometry description for the Laplacian on implicitly defined surfaces

- Mathematics
- 2018

In this paper, a high order finite element method for partial differential equations on smooth surfaces is proposed. The surface is defined as the intersection of a rectangular cuboid and an…

Non-degenerate Eulerian finite element method for solving PDEs on surfaces

- Mathematics, Physics
- 2013

Abstract The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in ℝN, N = 2; 3. The method builds upon the formulation introduced in [7], where a…

Stabilization of high order cut finite element methods on surfaces

- MathematicsIMA Journal of Numerical Analysis
- 2019

We develop and analyse a stabilization term for cut finite element approximations of an elliptic second-order partial differential equation on a surface embedded in ${\mathbb{R}}^d$. The new…

A Posteriori Error Estimates for Surface Finite Element Methods

- Mathematics
- 2014

OF DISSERTATION A POSTERIORI ERROR ESTIMATES FOR SURFACE FINITE ELEMENT METHODS Problems involving the solution of partial differential equations over surfaces appear in many engineering and…

## References

SHOWING 1-10 OF 27 REFERENCES

A finite element method for surface PDEs: matrix properties

- Computer Science, MathematicsNumerische Mathematik
- 2010

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

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2009

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

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2006

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

- 2007

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

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2009

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

- Mathematics
- 2007

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

- Mathematics
- 2010

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

- Mathematics
- 2009

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

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2007

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

- Computer ScienceSIAM J. Sci. Comput.
- 2005

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.