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

Abstract

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.

Bembel: The fast isogeometric boundary element C++ library for Laplace, Helmholtz, and electric wave equation

“…dat " ) ; 10 MaxwellSingle pde ( wavenumber ) ; 11 MatrixXd gridpoints = Util :: makeSph ereGrid (6 , 10) ; VectorXcd rhs = Rhs :: computeRhs ( disc , fun ) ; 20 HierarchicalMatrix < MaxwellSingle > hmat ( disc , m ul ti po l eD eg re e ) ; 21 GMRES < HierarchicalMatrix < MaxwellSingle > , IdentityPreconditioner > gmres ; 22 gmres . compute ( hmat ) ; 23 VectorXcd rho = gmres . solve ( rhs ) ; The convergence plots were computed around the pictured geometry, with a dipole in direction (0, 0, 1/100) placed at (1, 1, 0) , see [17] and the example programs.…”

“…The boundary element spaces for the Galerkin method are constructed as isogeometric multi-patch B-spline spaces, see [23]. Bembel uses equidistant knot vectors Ξ 1 = Ξ 2 with 2 L elements and the same polynomial degrees in each direction, i.e., p 1 = p 2 , on the unit square.…”

An Overview of Isogeometric Boundary Element Methods for Acoustic and Electromagnetic Scattering Problems

“…Let 0 < k < ∞ and let K be either R or C and p, i be integers with 0 ≤ p < k. As in [1,3], we define a p-open knot vector Ξ as a set of knots ξ i of the form…”

“…2.2]. As introduced in [3] and extended to multipatch domains in Ω and trace spaces on Γ = ∂Ω in [1], one can use the fact the derivative of a B-spline function is within the B-spline space of lower degree, to define a conforming discretisation of (1) in the patchwise reference domain given by…”

“…It can be shown that one can expect optimal convergence properties w.r.t. the norms of the corresponding space in (1) for multipatch domains, see [1].…”