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 straightforward generalisation to finite element methods

Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples

We present a new approach to three-dimensional electromagnetic scattering problems via fast isogeometric boundary element methods. Starting with an investigation of the theoretical setting around the electric field integral equation within the isogeometric framework, we show existence, uniqueness, and quasi-optimality of the isogeometric approach. For a fast and efficient computation, we then introduce and analyze an interpolation-based fast multipole method tailored to the isogeometric setting, which admits competitive algorithmic and complexity properties. This is followed by a series of numerical examples of industrial scope, together with a detailed presentation and interpretation of the results

Bounded commuting projections for multipatch spaces with non-matching interfaces

We present commuting projection operators on de Rham sequences of two-dimensional multipatch spaces with local tensor-product parametrization and non-matching interfaces. Our construction yields projection operators which are local and stable in any $L^p$ norm with $p \in [1,\infty]$: it applies to shape-regular spline patches with different mappings and local refinements, under the assumption that neighboring patches have nested resolutions and that interior vertices are shared by exactly four patches. It also applies to de Rham sequences with homogeneous boundary conditions. Following a broken-FEEC approach, we first consider tensor-product commuting projections on the single-patch de Rham sequences, and modify the resulting patch-wise operators so as to enforce their conformity and commutation with the global derivatives, while preserving their projection and stability properties with constants independent of both the diameter and inner resolution of the patches

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 straightforward generalisation to finite element methods