ABSTRACT. We present recent advances in domain decomposition methods for high-frequency time-harmonic wave problems, where subproblems of small sizes are solved using sparse direct solvers, and are combined using iterative techniques. We focus on a family of recently proposed quasi-optimal domain decomposition methods based on accurate approximations of the Dirichlet-to-Neumann map, combined with parallel, sweeping-type preconditioners.

ABSTRACT. We present a spectral numerical algorithm for the fast solution of elastodynamics problems in general 3D domains based on a FFT-speed Fourier Continuation (FC) approximation for accurate Fourier expansion of non-periodic functions. The high-order methodology yields physically correct solutions including those with traction conditions on curved boundaries. The approach is essentially without dispersion errors; entails mild CFL constraints; runs at a cost scaling linearly with discretization size; and can be efficiently parallelized for computing clusters.

ABSTRACT. We present new results on the existence of low-rank separable expansions for the Helmholtz fundamental solution when absorption is added to the wavenumber. Part of the motivation for these new results is to rigorously justify the choice of absorption added into sweeping-type preconditioners. (Recall that this idea of absorption comes from the so-called “Shifted-Laplacian” preconditioner.)

ABSTRACT. High order numerical methods exhibit dramatic gains in efficiency over low order methods by providing better accuracy on coarse grids, and therefore the computation time needed to obtain a desired level of accuracy in simulations is greatly reduced. In addition to the increased convergence rate, it has been shown that high order methods result in smaller dispersion errors than low order methods. In order to fit the needs of physical problems, high order methods must exhibit several capabilities, such as handling variable coefficient operators, realistic geometries, and different types of boundary conditions. We demonstrate a flexible approach that efficiently solves second order hyperbolic PDEs with high order accuracy through the combined methodology of compact high order finite differences and difference potentials.

ABSTRACT. We study numerically the e-rank of subblocks arising in Schur complement matrices of discretized three-dimensional Helmholtz problems. A small e-rank is the key ingredient for H-matrix techniques, and while Laplace-like problems have this property, the e-rank for the Helmholtz case is growing with increasing wave number. We study here the growth rate in the case of a heterogeneous Helmholtz problem with a checkerboard wave speed distribution, and compare it to the constant wave number case.

ABSTRACT. We formulate a finite element framework for the observation of surface plasmon-polaritons (SPP) on 2D materials, such as graphene. The 2D material is modeled as an idealized hypersurface with an effective complex-valued conductivity. The simulation of the scattering process uses a perfectly matched layer. A good resolution of the SPP structures on the hypersurface is achieved by using goal-oriented adaptive local mesh refinement utilizing the Dual-Weighted Residual (DWR) method.

ABSTRACT. We show how boundary integral equations on an uncertain boundary can be replaced by volume integral equations with a stochastic kernel but on a fixed support. This is advantageous for applications because with the volume integral equation, the Galerkin discretisation can be defined on a fixed domain instead of on (sample) realisations of the stochastic boundary.

ABSTRACT. Efficient implementations of high order discontinuous Galerkin (DG) methods on hexahedral meshes can incorporate both local variations in heterogeneous media using mass lumping techniques. However, because the extension of such techniques to simplicial elements is less straightforward, high order DG methods on triangular and tetrahedral meshes typically assume piecewise constant models of heterogeneous media. We present an alternative to mass-lumping techniques using weight-adjusted approximations to weighted L2 inner products, resulting in an energy stable, high order accurate, and low-storage method for acoustic and elastic wave propagation in arbitrary heterogeneous media and curvilinear meshes.

ABSTRACT. This work focuses on the well-posedness and stability of the linearised Euler equations with impedance boundary condition. The first part covers the acoustical case, where the complexity lies solely in the chosen impedance model. The existence of an asymptotically stable C0-semigroup of contractions is shown when the passive impedance admits a dissipative realisation; the only source of instability is the time-delay. The second part discusses the more challenging aeroacoustical case, which is the subject of ongoing research. A discontinuous Galerkin discretisation is used to investigate both cases.

ABSTRACT. Complex scaling is a popular method for treating scattering and resonance problems in open domains. For solving scattering problems it is common to use frequency dependent scaling parameters. Using similar ideas for resonance problems leads to non-linear eigenvalue problems. In this talk we analyze the discrete resonances of both, the frequency independent and the frequency dependent complex scaled Helmholtz equation.

ABSTRACT. We study elliptic boundary value problems where the coefficients are piecewise constant with respect to a partition of space into Lipschitz subdomains, focusing on the case of jumping coefficients arising in the principal part of the partial differential operator. We propose a boundary integral equation of the second kind posed on the interfaces of the partition, and involving only one unknown trace function of each interface. We provide a detailed analysis of the corresponding integral operator, proving well-posedness. We also present numerical results that exhibit a systematically stable condition number for the associated Galerkin matrices, so that GMRES seems to enjoy fast convergence independent of the mesh resolution.

ABSTRACT. We introduce a metric-based anisotropic mesh adaptation strategy for the fast multipole accelerated boundary element method (FM-BEM) applied to exterior boundary value problems of the three-dimensional Helmholtz equation. The present methodology is independent of discretization technique and iteratively constructs meshes refined in size, shape and orientation according to an “optimal” metric reliant on a reconstructed Hessian of the boundary solution. The resulting adaptation is anisotropic in nature and numerical examples demonstrate optimal convergence rates for domains that include geometric singularities such as corners and ridges.

ABSTRACT. We compare several approaches for handling the artificial outer boundaries that can be implemented with the standard FDTD method in 3D. Our goal is to obtain the asymptotic estimates of computational complexity for each class of methods and corroborate those with numerical results so as to show the advantages and disadvantages of the various methodologies.

ABSTRACT. The need for approximated solutions to a vector valued equation in electro-magnetic wave propagation raises the question of the amplitude, or polarization, of the approximated function. In this work we propose to incorporate geometric optics expansions into the design process of Generalized Plane Waves in order to take into account variable amplitudes.

ABSTRACT. We discuss the applicability of a time-reversal concept to the focusing of wave energy to multiple subsurface targets embedded within an arbitrarily heterogeneous three-dimensional elastic host. The motivation stems from an interest in facilitating oil ganglia mobility in support of enhanced oil recovery (EOR) methods. We quantify the focusing by a suitable motion metric, and provide numerical evidence supportive of the method's efficacy in illuminating the targets even when embedded within randomized media.

ABSTRACT. Complete radiation boundary conditions (CRBC) are local boundary condition sequences optimized for models related to the scalar wave equation, such as Maxwell's equations in nondispersive dielectrics. Here we consider their generalization to electromagnetic waves in dispersive media governed by Lorentz models. If only the permittivity is frequency-dependent, parameters can be chosen to guarantee rapid convergence with increasing order. For metamaterials, on the other hand, where both the permittivity and the permeability are frequency-dependent, so-called reverse modes exist and parameters for the standard CRBC cannot be chosen to guarantee convergence. Here we show how to modify the formulation so that convergence can be restored.

ABSTRACT. In this talk, we consider a nonlinear, dispersive, Maxwell model in its first order form, where the nonlinearity comes from the instantaneous electronic Kerr response together with the single resonance linear Lorentz dispersion. We design high order Discontinuous Galerkin (DG) discretizations in space for this model, and prove that the resulting semi-discrete and fully-discrete methods, based on leap-frog and implicit trapezoidal temporal schemes, are energy stable.

ABSTRACT. In this paper, we first represent the non-linear Hasegawa-Mima Partial Differential Equations (PDE's) as a coupled two linear Elliptic-Hyperbolic system of PDE's. We then apply the Petrov-Galerkin method to obtain a sequence of fixed-point approximate solutions that converge weakly to a solution of the Hasegawa-Mima problem that is simulated using a Finite Element method.