Multipatch approximation of the de Rham sequence and its traces in isogeometric analysis

We define a conforming B-spline discretisation of the de Rham complex on multipatch geometries. We introduce and analyse the properties of interpolation operators onto these spaces which commute w.r.t. the surface differential operators. Using these results as a basis, we derive new convergence results of optimal order w.r.t. the respective energy spaces and provide approximation properties of the spline discretisations of trace spaces for application in the theory of isogeometric boundary element methods. Our analysis allows for a straight forward generalisation to finite element methods.

[1]  R. Hiptmair,et al.  Galerkin Boundary Element Methods for Electromagnetic Scattering , 2003 .

[2]  W. Hackbusch,et al.  H 2 -matrix approximation of integral operators by interpolation , 2002 .

[3]  A. Bossavit.  Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements , 1997 .

[4]  J. Wloka,et al.  Partial differential equations: Strongly elliptic differential operators and the method of variations , 1987 .

[5]  Thomas-Peter Fries,et al.  Isogeometric Boundary Element analysis with elasto-plastic inclusions. Part 1: Plane problems , 2016 .

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

[7]  A. Bossavit.  Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism , 1988 .

[8]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[9]  T. Rabczuk,et al.  A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis , 2012 .

[10]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[11]  Giancarlo Sangalli,et al.  Approximation estimates for isogeometric spaces in multipatch geometries , 2015 .

[12]  Andrew F. Peterson.  Mapped Vector Basis Functions for Electromagnetic Integral Equations , 2006, Mapped Vector Basis Functions for Electromagnetic Integral Equations.

[13]  C. Schwab,et al.  Boundary Element Methods , 2010 .

[14]  Marc Gerritsma,et al.  Edge Functions for Spectral Element Methods , 2011 .

[15]  M. Costabel,et al.  Singularities of Electromagnetic Fields¶in Polyhedral Domains , 2000 .

[16]  Martin Costabel,et al.  Polynomial extension operators for H1, H(curl) and H(div)-spaces on a cube , 2008, Math. Comput..

[17]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[18]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

[19]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[20]  W. D’haeseleer,et al.  Flux Coordinates and Magnetic Field Structure , 1991 .

[21]  Helmut Harbrecht,et al.  Comparison of fast boundary element methods on parametric surfaces , 2013 .

[22]  Lucy Weggler.  High order boundary element methods , 2011 .

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

[24]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[25]  E. Valdinoci,et al.  Hitchhiker's guide to the fractional Sobolev spaces , 2011, 1104.4345.

[26]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[27]  Olaf Steinbach,et al.  Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements , 2007 .

[28]  Peter Monk,et al.  Finite Element Methods for Maxwell's Equations , 2003 .

[29]  Maharavo Randrianarivony,et al.  From Computer Aided Design to wavelet BEM , 2009, Comput. Vis. Sci..

[30]  Les A. Piegl,et al.  The NURBS book (2nd ed.) , 1997 .

[31]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[32]  Giancarlo Sangalli,et al.  Isogeometric Discrete Differential Forms in Three Dimensions , 2011, SIAM J. Numer. Anal..

[33]  Alfio Quarteroni,et al.  Domain Decomposition Methods for Partial Differential Equations , 1999 .

[34]  E. T. Y. Lee,et al.  Marsden's identity , 1996, Comput. Aided Geom. Des..

[35]  Ludmil T. Zikatanov,et al.  Some observations on Babu\vs}ka and Brezzi theories , 2003, Numerische Mathematik.

[36]  Stefan Kurz,et al.  Fast Boundary Element Methods in Computational Electromagnetism , 2007 .

[37]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[38]  Helmut Harbrecht,et al.  An interpolation‐based fast multipole method for higher‐order boundary elements on parametric surfaces , 2016 .

[39]  Snorre H. Christiansen,et al.  The electric field integral equation on Lipschitz screens: definitions and numerical approximation , 2003, Numerische Mathematik.

[40]  G. Sangalli,et al.  Isogeometric analysis in electromagnetics: B-splines approximation , 2010 .

[41]  Stefan Kurz,et al.  A fast isogeometric BEM for the three dimensional Laplace- and Helmholtz problems , 2017, 1708.09162.

[42]  Zeger Bontinck,et al.  Recent Advances of Isogeometric Analysis in Computational Electromagnetics , 2017, ArXiv.

[43]  Rafael Vázquez Hernández,et al.  An isogeometric boundary element method for electromagnetic scattering with compatible B-spline discretizations , 2017, J. Comput. Phys..

[44]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[45]  Giancarlo Sangalli,et al.  Mathematical analysis of variational isogeometric methods* , 2014, Acta Numerica.

[46]  Thomas-Peter Fries,et al.  Fast Isogeometric Boundary Element Method based on Independent Field Approximation , 2014, ArXiv.

[47]  Patrick Ciarlet,et al.  On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications , 2001 .

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

[49]  A. Bu.  On traces for functional spaces related to Maxwell's equations Part I: An integration by parts formula in Lipschitz polyhedra , 2001 .

[50]  Peter Monk,et al.  An analysis of Ne´de´lec's method for the spatial discretization of Maxwell's equations , 1993 .

[51]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.