The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems

Abstract We propose a new finite element method for embedded domain computations, which falls in the category of surrogate/approximate boundary algorithms. The key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary, and to appropriately modify the shifted boundary conditions, enforced weakly, in order to preserve optimal convergence rates of the numerical solution. This process yields a method which, in our view, is simple, efficient, and also robust, since it is not affected by the small-cut-cell problem. Although general in nature, here we apply this new concept to the Poisson and Stokes problems. We present in particular the full analysis of stability and convergence for the case of the Poisson operator, and numerical tests for both the Poisson and Stokes equations, for geometries of progressively higher complexity.

[1]  Rémi Abgrall,et al.  Computations of compressible multifluids , 2001 .

[2]  Fehmi Cirak,et al.  A fixed‐grid b‐spline finite element technique for fluid–structure interaction , 2014 .

[3]  Randall J. LeVeque,et al.  Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension , 1997, SIAM J. Sci. Comput..

[4]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[5]  Ulrich Reif,et al.  Weighted Extended B-Spline Approximation of Dirichlet Problems , 2001, SIAM J. Numer. Anal..

[6]  Boyce E. Griffith,et al.  An adaptive, formally second order accurate version of the immersed boundary method , 2007, J. Comput. Phys..

[7]  Isaac Harari,et al.  An efficient finite element method for embedded interface problems , 2009 .

[8]  Jinchao Xu,et al.  Regularity estimates for elliptic boundary value problems in Besov spaces , 2003, Math. Comput..

[9]  J. Douglas,et al.  Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods , 1976 .

[10]  Kevin G. Wang,et al.  Predictive Simulation of Underwater Implosion: Coupling Multi-Material Compressible Fluids With Cracking Structures , 2014 .

[11]  W. Wall,et al.  A face‐oriented stabilized Nitsche‐type extended variational multiscale method for incompressible two‐phase flow , 2015 .

[12]  Chandrasekhar Annavarapu,et al.  An efficient finite element method for interface problems , 2013 .

[13]  L. Heltai,et al.  Stability Results and Algorithmic Strategies for the Finite Element Approach to the Immersed Boundary Method , 2006 .

[14]  Benedikt Schott,et al.  A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier–Stokes equations , 2014 .

[15]  Charbel Farhat,et al.  A higher-order generalized ghost fluid method for the poor for the three-dimensional two-phase flow computation of underwater implosions , 2008, J. Comput. Phys..

[16]  Fehmi Cirak,et al.  Subdivision-stabilised immersed b-spline finite elements for moving boundary flows , 2012 .

[17]  Peter Hansbo,et al.  Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem , 2014 .

[18]  Isaac Harari,et al.  A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces , 2007 .

[19]  John E. Dolbow,et al.  A robust Nitsche’s formulation for interface problems , 2012 .

[20]  Ernst Rank,et al.  The finite cell method for three-dimensional problems of solid mechanics , 2008 .

[21]  Boyce E. Griffith,et al.  On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems , 2005 .

[22]  Stephen Childress,et al.  An Introduction to Theoretical Fluid Mechanics , 2009 .

[23]  K. Höllig.  Finite element methods with B-splines , 1987 .

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

[25]  Thomas J. R. Hughes,et al.  The Stokes problem with various well-posed boundary conditions - Symmetric formulations that converge for all velocity/pressure spaces , 1987 .

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

[27]  L. Heltai,et al.  Mathematical Models and Methods in Applied Sciences Vol. 17, No. 10 (17) 1479–1505 c ○ World Scientific Publishing Company NUMERICAL STABILITY OF THE FINITE ELEMENT IMMERSED BOUNDARY METHOD , 2005 .

[28]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[29]  H. Fasel,et al.  A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains , 2005 .

[30]  Ernst Rank,et al.  Finite cell method , 2007 .

[31]  Charbel Farhat,et al.  Computational algorithms for tracking dynamic fluid–structure interfaces in embedded boundary methods , 2012 .

[32]  C. Peskin.  Flow patterns around heart valves: A numerical method , 1972 .

[33]  Xianyi Zeng,et al.  An enhanced FIVER method for multi-material flow problems with second-order convergence rate , 2017, J. Comput. Phys..

[34]  Parviz Moin,et al.  Prediction of wall-pressure fluctuation in turbulent flows with an immersed boundary method , 2009, J. Comput. Phys..

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

[36]  L. Heltai,et al.  A finite element approach to the immersed boundary method , 2003 .

[37]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..