Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge–Kutta convolution quadrature

Ballani, Jonas; Banjai, Lehel; Sauter, Stéfan A.; Veit, Alexander

doi:10.1007/s00211-012-0503-7

Cited by 25 publications

(41 citation statements)

References 36 publications

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“…In [9, Section 5] the continuity of these operators was proven, without giving an explicit dependence on s. Such bounds are crucial in the analysis later, therefore we now show s-explicit estimates for the boundary integral operators. Our result is based on [2]. Furthermore, using the potential representation of the solution (2.5), the averages of the traces can be expressed using the operators V and K in the following way:…”

Section: Boundary Integral Operatorsmentioning

confidence: 99%

“…In Section 2 we recapitulate the basic theory for Maxwell's equation in the Laplace domain. Based on Buffa and Hiptmair [9], and further on [10,2], we describe the right boundary space, which allows for a rigorous boundary integral formulation for Maxwell's equations. Then the boundary integral operators are obtained in a usual way from the single and double layer potentials.…”

Section: Introductionmentioning

confidence: 99%

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Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations

Kovács

Lubich

2017

Numer. Math.

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Maxwell's equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical method only involves the interior domain and its boundary. The transparent boundary conditions are imposed via a time-dependent boundary integral operator that is shown to satisfy a coercivity property. The stability of the numerical method relies on this coercivity and on an antisymmetric structure of the discretized equations that is inherited from a weak first-order formulation of the continuous equations. The method proposed here uses a discontinuous Galerkin method and the leapfrog scheme in the interior and is coupled to boundary elements and convolution quadrature on the boundary. The method is explicit in the interior and implicit on the boundary. Stability and convergence of the spatial semidiscretization are proven, and with a computationally simple stabilization term, this is also shown for the full discretization.

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Section: Boundary Integral Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations

Kovács

Lubich

2017

Numer. Math.

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“…We envision our results to be a unifying and instrumental step in carrying out the analysis of the full discretization for different time semidiscretization strategies like finite differences, space-time Galerkin or convolution quadrature. It is also surprising (although this is not new [1,9]) to note that the TDEFIE is amenable to a general Galerkin discretization-in-space, and no discrete Hodge decompositions are needed, as opposed to the requirements of the frequency domain EFIE [13].…”

Section: Introductionmentioning

confidence: 99%

New mapping properties of the time domain electric field integral equation

Qiu

Sayas

2016

ESAIM: M2AN

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We show some improved mapping properties of the Time Domain Electric Field Integral Equation and of its Galerkin semidiscretization in space. We relate the weak distributional framework with a stronger class of solutions using a group of strongly continuous operators. The stability and error estimates we derive are sharper than those in the literature.Proof. This is a straightforward consequence of the definition of G h and of the characterization of the Maxwell single layer potential by the transmission problem (3.2). Note that the second equation in (3.9) can be equivalently written as a Galerkin semidiscreteProposition 3.4. Let J ∈ TD(H −1/2 (div Γ , Γ)) and define

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“…In addition, the use of implicit Runge–Kutta integration techniques is now well‐understood (for simplicity, we shall not use Runge–Kutta methods in this article). There has been significant progress on extending the analysis of the method to more general boundary conditions and some progress in analyzing electromagnetic problems . Of special importance to us is the work of Banjai and Sauter who show how to compute convolution quadrature solutions via the solution of several Laplace domain problems and an inverse transform .…”

Section: Introductionmentioning

confidence: 99%

The time‐domain Lippmann–Schwinger equation and convolution quadrature

Lechleiter

Monk

2014

Numerical Methods Partial

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We consider time‐domain acoustic scattering from a penetrable medium with a variable sound speed. This problem can be reduced to solve a time‐domain volume Lippmann–Schwinger integral equation. Using convolution quadrature in time and trigonometric collocation in space, we can compute an approximate solution. We prove that the time‐domain Lippmann–Schwinger equation has a unique solution and prove conditional convergence and error estimates for the fully discrete solution for globally smooth sound speeds. Preliminary numerical results show that the method behaves well even for discontinuous sound speeds. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 517–540, 2015

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Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge–Kutta convolution quadrature

Cited by 25 publications

References 36 publications

Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations

Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations

New mapping properties of the time domain electric field integral equation

The time‐domain Lippmann–Schwinger equation and convolution quadrature

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