A Nitsche-based domain decomposition method for hypersingular integral equations

We introduce and analyze a Nitsche-based domain decomposition method for the solution of hypersingular integral equations. This method allows for discretizations with non-matching grids without the necessity of a Lagrangian multiplier, as opposed to the traditional mortar method. We prove its almost quasi-optimal convergence and underline the theory by a numerical experiment.

[1]  Miguel A. Fernández,et al.  Robin Based Semi-Implicit Coupling in Fluid-Structure Interaction: Stability Analysis and Numerics , 2009, SIAM J. Sci. Comput..

[2]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[3]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[4]  Faker Ben Belgacem,et al.  The Mortar finite element method with Lagrange multipliers , 1999, Numerische Mathematik.

[5]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[6]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[7]  P. Hansbo,et al.  A finite element method for domain decomposition with non-matching grids , 2003 .

[8]  A. Buffa,et al.  On traces for H(curl,Ω) in Lipschitz domains , 2002 .

[9]  Clemens Pechstein,et al.  Boundary element tearing and interconnecting methods in unbounded domains , 2009 .

[10]  Norbert Heuer,et al.  Mortar Boundary Elements , 2010, SIAM J. Numer. Anal..

[11]  Ernst P. Stephan,et al.  Boundary integral equations for screen problems in IR3 , 1987 .

[12]  J. Nédélec,et al.  Integral equations with non integrable kernels , 1982 .

[13]  A. Maue,et al.  Zur Formulierung eines allgemeinen Beugungs-problems durch eine Integralgleichung , 1949 .

[14]  B. Wohlmuth,et al.  A comparison of mortar and Nitsche techniques for linear elasticity , 2004 .

[15]  A. Reusken,et al.  Nitsche’s Method for a Transport Problem in Two-phase Incompressible Flows , 2009 .

[16]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[17]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[18]  Norbert Heuer,et al.  The boundary element method with Lagrangian multipliers , 2009 .

[19]  P. Hansbo,et al.  Nitsche's method combined with space–time finite elements for ALE fluid–structure interaction problems☆ , 2004 .

[20]  Helio J. C. Barbosa,et al.  Boundary Lagrange multipliers in finite element methods: Error analysis in natural norms , 1992 .

[21]  Norbert Heuer,et al.  Crouzeix–Raviart boundary elements , 2009, Numerische Mathematik.

[22]  Rolf Stenberg,et al.  On some techniques for approximating boundary conditions in the finite element method , 1995 .

[23]  P. Hansbo,et al.  A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity , 2009 .

[24]  Erik Burman,et al.  Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility , 2009 .

[25]  Norbert Heuer,et al.  Additive Schwarz method for the p-version of the boundary element method for the single layer potential operator on a plane screen , 2001, Numerische Mathematik.

[26]  Ernst P. Stephan,et al.  A boundary integral equation method for three‐dimensional crack problems in elasticity , 1986 .

[27]  Olaf Steinbach,et al.  Inexact Fast Multipole Boundary Element Tearing and Interconnecting Methods , 2007 .

[28]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[29]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[30]  Houde Han,et al.  The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity , 1994 .

[31]  Thanh Tran,et al.  Additive schwarz methods for the H-version boundary element method , 1996 .

[32]  Marc Garbey,et al.  Asymptotic and numerical methods for partial differential equations with critical parameters , 1993 .

[33]  Norbert Heuer,et al.  On the h-p version of the boundary element method for Symm's integral equation on polygons , 1993 .

[34]  Ernst P. Stephan,et al.  Two-level methods for the single layer potential in ℝ3 , 1998, Computing.

[35]  Helio J. C. Barbosa,et al.  The finite element method with Lagrange multiplier on the boundary: circumventing the Babuscka-Brezzi condition , 1991 .

[36]  Martin Costabel,et al.  Boundary Integral Operators on Lipschitz Domains: Elementary Results , 1988 .

[37]  Anthony T. Patera,et al.  Domain Decomposition by the Mortar Element Method , 1993 .