Analysis of a High-Order Trace Finite Element Method for PDEs on Level Set Surfaces

We present a new high-order finite element method for the discretization of partial differential equations on stationary smooth surfaces which are implicitly described as the zero level of a level set function. The discretization is based on a trace finite element technique. The higher discretization accuracy is obtained by using an isoparametric mapping of the volume mesh, based on the level set function, as introduced in [C. Lehrenfeld, Comp. Meth. Appl. Mech. Engrg., 300 (2016), pp. 716--733]. The resulting trace finite element method is easy to implement. We present an error analysis of this method and derive optimal order $H^1(\Gamma)$-norm error bounds. A second topic of this paper is a unified analysis of several stabilization methods for trace finite element methods. Only a stabilization method which is based on adding an anisotropic diffusion in the volume mesh is able to control the condition number of the stiffness matrix also for the case of higher-order discretizations. Results of numerical e...

[1]  Arnold Reusken,et al.  Analysis of trace finite element methods for surface partial differential equations , 2015 .

[2]  Andreas Dedner,et al.  High Order Discontinuous Galerkin Methods for Elliptic Problems on Surfaces , 2015, SIAM J. Numer. Anal..

[3]  Alan Demlow,et al.  An Adaptive Finite Element Method for the Laplace-Beltrami Operator on Implicitly Defined Surfaces , 2007, SIAM J. Numer. Anal..

[4]  Jörg Grande,et al.  Eulerian Finite Element Methods for Parabolic Equations on Moving Surfaces , 2014, SIAM J. Sci. Comput..

[5]  Arnold Reusken,et al.  A Higher Order Finite Element Method for Partial Differential Equations on Surfaces , 2016, SIAM J. Numer. Anal..

[6]  Maxim A. Olshanskii,et al.  A Finite Element Method for Elliptic Equations on Surfaces , 2009, SIAM J. Numer. Anal..

[7]  Maxim A. Olshanskii,et al.  An Eulerian Space-Time Finite Element Method for Diffusion Problems on Evolving Surfaces , 2013, SIAM J. Numer. Anal..

[8]  Christoph Lehrenfeld,et al.  High order unfitted finite element methods on level set domains using isoparametric mappings , 2015, ArXiv.

[9]  Peter Hansbo,et al.  Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions , 2016, ESAIM: Mathematical Modelling and Numerical Analysis.

[10]  Ulrich Langer,et al.  Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces , 2016 .

[11]  P. Hansbo,et al.  An unfitted finite element method, based on Nitsche's method, for elliptic interface problems , 2002 .

[12]  Maxim A. Olshanskii,et al.  A finite element method for surface PDEs: matrix properties , 2009, Numerische Mathematik.

[13]  P. Hansbo,et al.  Full gradient stabilized cut finite element methods for surface partial differential equations , 2016, 1602.01512.

[14]  Maxim A. Olshanskii,et al.  A stabilized finite element method for advection-diffusion equations on surfaces , 2013, 1301.3741.

[15]  Peter Hansbo,et al.  A cut discontinuous Galerkin method for the Laplace-Beltrami operator , 2015, 1507.05835.

[16]  Charles M. Elliott,et al.  Unfitted Finite Element Methods Using Bulk Meshes for Surface Partial Differential Equations , 2013, SIAM J. Numer. Anal..

[17]  Maxim A. Olshanskii,et al.  Error Analysis of a Space-Time Finite Element Method for Solving PDEs on Evolving Surfaces , 2014, SIAM J. Numer. Anal..

[18]  M. Lenoir.  Optimal isoparametric finite elements and error estimates for domains involving curved boundaries , 1986 .

[19]  Alan Demlow,et al.  Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces , 2009, SIAM J. Numer. Anal..

[20]  Maxim A. Olshanskii,et al.  A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS ∗ , 2014, 1406.7694.

[21]  Jean-Luc Guermond,et al.  Finite element quasi-interpolation and best approximation , 2015, 1505.06931.

[22]  Christoph Lehrenfeld,et al.  Analysis of a high order unfitted finite element method for elliptic interface problems , 2016, 1602.02970.

[23]  Arnold Reusken,et al.  A space-time FEM for PDEs on evolving surfaces , 2014, 1403.0277.

[24]  Peter Hansbo,et al.  A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator , 2013, 1312.1097.