The aggregated unfitted finite element method for elliptic problems

Abstract Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hindersthe practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.

[1]  Randall J. LeVeque,et al.  A High-Resolution Rotated Grid Method for Conservation Laws with Embedded Geometries , 2005, SIAM J. Sci. Comput..

[2]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[3]  Juan José Ródenas,et al.  Exact 3D boundary representation in finite element analysis based on Cartesian grids independent of the geometry , 2015 .

[4]  S. Agmon.  Lectures on Elliptic Boundary Value Problems , 1965 .

[5]  S. Bordas,et al.  A robust preconditioning technique for the extended finite element method , 2011 .

[6]  Yuri Bazilevs,et al.  An immersogeometric variational framework for fluid-structure interaction: application to bioprosthetic heart valves. , 2015, Computer methods in applied mechanics and engineering.

[7]  Dominik Schillinger,et al.  The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models , 2015 .

[8]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

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

[10]  Peter Hansbo,et al.  CutFEM: Discretizing geometry and partial differential equations , 2015 .

[11]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[12]  Santiago Badia,et al.  FEMPAR: An Object-Oriented Parallel Finite Element Framework , 2017, Archives of Computational Methods in Engineering.

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

[14]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

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

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

[17]  F. de Prenter,et al.  Condition number analysis and preconditioning of the finite cell method , 2016, 1601.05129.

[18]  P. Hansbo,et al.  Fictitious domain finite element methods using cut elements , 2012 .

[19]  R. Tuminaro,et al.  Inexact Schwarz‐algebraic multigrid preconditioners for crack problems modeled by extended finite element methods , 2012 .

[20]  Paul Houston,et al.  Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems , 2001, SIAM J. Numer. Anal..

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

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

[23]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[24]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[25]  David E. Keyes,et al.  A Quasi-algebraic Multigrid Approach to Fracture Problems Based on Extended Finite Elements , 2012, SIAM J. Sci. Comput..

[26]  Cv Clemens Verhoosel,et al.  Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone , 2015 .

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

[28]  Mats G. Larson,et al.  A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary , 2013, Numerische Mathematik.

[29]  Roland Becker,et al.  Mesh adaptation for Dirichlet flow control via Nitsche's method , 2002 .

[30]  Wolfgang A. Wall,et al.  An accurate, robust, and easy-to-implement method for integration over arbitrary polyhedra: Application to embedded interface methods , 2014, J. Comput. Phys..

[31]  Santiago Badia,et al.  Multilevel Balancing Domain Decomposition at Extreme Scales , 2016, SIAM J. Sci. Comput..

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

[33]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[34]  F. Kummer.  Extended discontinuous Galerkin methods for two‐phase flows: the spatial discretization , 2017 .

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

[36]  William L. Briggs,et al.  A multigrid tutorial, Second Edition , 2000 .

[37]  Santiago Badia,et al.  Robust and scalable domain decomposition solvers for unfitted finite element methods , 2018, J. Comput. Appl. Math..

[38]  Fabio Nobile,et al.  Fluid-structure partitioned procedures based on Robin transmission conditions , 2008, J. Comput. Phys..