A uniformly well-conditioned, unfitted Nitsche method for interface problems
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A uniformly well-conditioned, unfitted Nitsche method for interface problems Eddie Wadbro · Sara Zahedi · Gunilla Kreiss · Martin Berggren
Received: 7 July 2012 / Accepted: 20 December 2012 / Published online: 23 January 2013 © Springer Science+Business Media Dordrecht 2013
Abstract A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented. The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and jump conditions at the interface are weakly enforced using a variant of Nitsche’s method. In our method, the solutions on each side of the interface are extended to the entire domain which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. We prove that the method provides optimal convergence order in the energy and the L2 norms and a condition number of the system matrix that is independent of the position of the interface relative to the mesh. Numerical experiments confirm the theoretical results and demonstrate optimal convergence order also for the pointwise errors. Keywords Interface problem · Nitsche’s method · Interior penalties · Finite element methods Mathematics Subject Classification (2000) 35J05 · 35J20 · 65N15 · 65N30 Communicated by Ragnar Winther. E. Wadbro () · M. Berggren Department of Computing Science, Umeå University, 901 87 Umeå, Sweden e-mail: [email protected] M. Berggren e-mail: [email protected] S. Zahedi · G. Kreiss Department of Information Technology, Uppsala University, Box 337, 751 05 Uppsala, Sweden S. Zahedi e-mail: [email protected] G. Kreiss e-mail: [email protected]
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1 Introduction There is a growing interest in accurate and efficient numerical methods for problems in which the solution exhibits strong or weak discontinuities—that is, jumps in the solution or its derivatives—along a surface in the interior of the solution domain. The application we have in mind is two-phase flow where differences in fluid viscosity and the surface-tension force give rise to a weak discontinuity in the velocity field and a strong discontinuity in the pressure. The weak and strong discontinuities are for two-phase flows typically formulated as jump conditions for the normal stress at the interface [16]. Weak discontinuities in the solution of a partial differential equation can be approximated with optimal order by globally continuous finite element functions, provided that the interface coincides with mesh lines [2, 7]. The mortar method introduces Lagrange multipliers supported on the interface to enforce weak discontinuities, allowing mesh vertices to be non-matching at the interface without loss of accuracy [14, 17]. The interface still needs to be aligned with mesh lines; otherwise, the approximation order will be ruined. The requirement that the mesh should conform to the interface leads t
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