Coupling FEM with a Multiple-Subdomain Trefftz Method

Casati, Daniele; Hiptmair, Ralf

doi:10.1007/s10915-020-01179-z

Cited by 3 publications

(7 citation statements)

References 26 publications

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“…Finally, [27] assumes that equation parameters are piecewise constant in the Trefftz domain, which induces a partition: one unbounded subdomain and other bounded, but possibly very large, subdomains, each requiring its own Trefftz discretization space. Hence, coupling strategies are extended to handle more than one Trefftz domain for the case of 2D Helmholtz equation [64,p.…”

Section: Scopementioning

confidence: 99%

Coupling finite elements and auxiliary sources

Casati¹,

Hiptmair²

2019

Computers & Mathematics with Applications

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We propose a hybrid method to solve time-harmonic Maxwell's equations in R 3 through the Finite Element Method (FEM) in a bounded region encompassing parameter inhomogeneities, coupled with the Multiple Multipole Program (MMP) in the unbounded complement.FEM and MMP enjoy complementary capabilities. FEM requires a mesh of the computational domain of interest. This is expensive, but can treat inhomogeneous materials, shapes with corners, or other singularities. Moreover, FEM allows a purely local construction of the discrete system of equations.On the other hand, MMP belongs to the class of methods of auxiliary sources and of Trefftz methods, as it employs point sources that are exact solutions of the homogeneous equations as (global) basis functions: one only needs to evaluate them on hypersurfaces to obtain the discrete problem, and the resulting linear combination is valid in the whole domain where the equations hold, which can be unbounded. MMP performs well where the electromagnetic field is smooth, i.e. in the free space far from physical sources and material interfaces.Thus, a natural way to combine the strengths of these methods arises when one needs to simulate the electromagnetic field of components with inhomogeneous parameters or nonsmooth shapes surrounded by free space: use FEM on a mesh defined on the components and MMP in the unbounded domain outside. The boundary between the FEM and MMP domains can be nonphysical if one surrounds the components by a conforming mesh of an "airbox" also modeled by FEM.The interface conditions on the surface of the FEM domain are key to accurate coupled FEM-MMP solutions. Integrating by parts the variational form solved by FEM, surface integrals appear, through which one can impose interface conditions by substituting the ansatz of MMP.iii However, one interface condition (for example, the continuity of the tangential trace for Maxwell's equations) cannot be imposed in this way.To include this additional condition, we derive four methods from Lagrangian functionals that enforce both the variational form in the FEM domain and all the interface conditions between different discretizations:1. Least-squares-based coupling using techniques from PDE-constrained optimization. This approach optimizes a functional for the additional condition, subject to a constraint expressed by the variational form of FEM.2. Discontinuous Galerkin (DG) between the meshed FEM domain and the single-entity "MMP mesh", interpreting MMP as FEM with special basis functions.3. Multi-field variational formulation in the spirit of mortar finite element methods, relying on the same interpretation of MMP as the DG-based coupling.4. Coupling through the Dirichlet-to-Neumann operator : here we introduce a weak formulation of the additional condition where MMP basis functions are used as test functions. This approach is generalized to couple MMP with any numerical method based on volume meshes that fits co-chain calculus.Furthermore, to minimize the meshed region and for the sake of generalization, we as...

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Section: Scopementioning

confidence: 99%

Coupling finite elements and auxiliary sources

Casati¹,

Hiptmair²

2019

Computers & Mathematics with Applications

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“…22,23 Essentially, the Mie-Vekua approach expands some scalar field in a 2D multiply-connected domain by a multipole expansion supplemented with generalized harmonic polynomials. Extending these ideas, MMP introduces more basis functions (multiple multipoles) than required according to Vekua's theory 23 to span the MMP Trefftz spaces (7).…”

Section: Multiple Multipole Programmentioning

confidence: 99%

“…Finally, Reference 7 assumes that material parameters are piecewise constant in the Trefftz domain, similarly to what is done in this work, to solve the 2D Helmholtz equation. The approaches we propose here to realize the coupling between FEM and more than one Trefftz domain have been described there for the first time.…”

Section: Introductionmentioning

confidence: 99%

“…The novelty of the present work lies in using FEM with more than one Trefftz domain to solve nontrivial problems involving time‐harmonic Maxwell's equations, while Reference 7 is confined to 2D Helmholtz. This work also includes a numerical example involving infinite layered media 12 (section 1.2.2).…”

Section: Introductionmentioning

confidence: 99%

“…6 Here, Ω 0 m does not fit into the partition of Section 1.2, where we require that (μ, ɛ) ∈ ℂ 2 is constant in Ω 0 m . 7 Hence, the PDE-constrained coupling doubles the number of FEM degrees of freedom. Conversely, the number of degrees of freedom of all the other coupling approaches is approximately the same as standard FEM.…”

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Coupling finite elements and auxiliary sources for electromagnetic wave propagation

Casati

Hiptmair

Smajić

2020

Int J Numerical Modelling

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We propose four approaches to solve time-harmonic Maxwell's equations in R 3 through the finite element method (FEM) in a bounded region encompassing parameter inhomogeneities, coupled with the multiple multipole program (MMP) in the unbounded complement. MMP belongs to the class of methods of auxiliary sources and of Trefftz methods, as it employs point sources that spawn exact solutions of the homogeneous equations. Each of these sources is anchored at a point that, if singular, is placed outside the respective domain of approximation. In the MMP domain, we assume that material parameters are piecewise constant, which induces a partition into one unbounded subdomain and other bounded, but possibly very large, subdomains, each requiring its own MMP trial space. Hence, in addition to the transmission conditions between the FEM and MMP domains, one also has to impose transmission conditions connecting the MMP subdomains. Coupling approaches arise from seeking stationary points of Lagrangian functionals that both enforce the variational form of the equations in the FEM domain and match the different trial functions across subdomain interfaces. We discuss the following approaches: 1 Least-squares-based coupling using techniques from PDE-constrained optimization. 2 Discontinuous Galerkin coupling between the meshed FEM domain and the single-entity MMP subdomains. 3 Multi-field variational formulation in the spirit of mortar FEMs. 4 Coupling through the Dirichlet-to-Neumann operator. We compare these approaches in a series of numerical experiments with different geometries and material parameters, including examples that exhibit triple-point singularities and infinite layered media.

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Trefftz co-chain calculus

Casati

Codecasa

Hiptmair

et al. 2022

Z. Angew. Math. Phys.

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We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on $${\mathbb {R}}^n$$ R n . In the spirit of domain decomposition, we partition $${\mathbb {R}}^n=\Omega \cup \Gamma \cup \Omega _{+}$$ R n = Ω ∪ Γ ∪ Ω + , $$\Omega $$ Ω a bounded Lipschitz polyhedron, $$\Gamma :=\partial \Omega $$ Γ : = ∂ Ω , and $$\Omega _{+}$$ Ω + unbounded. In $$\Omega $$ Ω , we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In $$\Omega _{+}$$ Ω + , we rely on a meshless Trefftz–Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across $$\Gamma $$ Γ . Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results.

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Coupling FEM with a Multiple-Subdomain Trefftz Method

Cited by 3 publications

References 26 publications

Coupling finite elements and auxiliary sources

Coupling finite elements and auxiliary sources

Coupling finite elements and auxiliary sources for electromagnetic wave propagation

Trefftz co-chain calculus

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