In this study, we present spectral enclosures and accumulation of eigenvalues of a class of operator functions with several unbounded operator coefficients. Our findings have direct relevance to the third-order Moore-Gibson-Thompson equation with memory and additional damping. The new results include sufficient conditions for the accumulation of branches of eigenvalues to the essential spectrum and new spectral enclosures for operator functions with several unbounded operator coefficients. To illustrate the analytical results, we apply the abstract findings to concrete equations of the Moore-Gibson-Thompson type. Additionally, we employ numerical computations to further elucidate the analytical results.

In this paper, we present an adaptive spectral projection based finite element method to numerically approximate the solution of the wave equation with memory. The adaptivity is not restricted to the mesh (-adaptivity), but it is also applied to the size of the computed spectrum (-adaptivity). The meshes are refined using a residual based error estimator, while the size of the computed spectrum is adapted using the  norm of the error of the projected data. We show that the approach can be very efficient and accurate.

In this paper, we consider the numerical inverse Laplace transform for distributed order time-fractional equations, where a discontinuous Galerkin scheme is used to discretize the problem in space. The success of Talbot’s approach for the computation of the inverse Laplace transform depends critically on the problem’s spectral properties and we present a method to numerically enclose the spectrum and compute resolvent estimates independent of the problem size. The new results are applied to time-fractional wave and diffusion-wave equations of distributed order.

In the quest for the development of faster and more reliable technologies, the abilityto control the propagation, confinement, and emission of light has become crucial. Thedesign of guide mode resonators and perfect absorbers has proven to be of fundamentalimportance. In this project, we consider the shape optimization of a periodic dielectricslab aiming at efficient directional routing of light to reproduce similar features of a guidemode resonator. For this, the design objective is to maximize the routing efficiency of anincoming wave. That is, the goal is to promote wave propagation along the periodic slab.A Helmholtz problem with a piecewise constant and periodic refractive index mediummodels the wave propagation, and an accurate Robin-to-Robin map models an exteriordomain. We propose an optimal design strategy that consists of representing the dielectricinterface by a finite Fourier formula and using its coefficients as the design variables.Moreover, we use a high order finite element (FE) discretization combined with a bilinearTransfinite Interpolation formula. This setting admits explicit differentiation with respectto the design variables, from where an exact discrete adjoint method computes thesensitivities. We show in detail how the sensitivities are obtained in the quasi-periodicdiscrete setting. The design strategy employs gradient-based numerical optimization, whichconsists of a BFGS quasi-Newton method with backtracking line search. As a test caseexample, we present results for the optimization of a so-called single port perfect absorber.We test our strategy for a variety of incoming wave angles and different polarizations. Inall cases, we efficiently reach designs featuring high routing efficiencies that satisfy therequired criteria.

In this paper, we compare two approaches to numerically approximate the solution of second-order Gurtin-Pipkin type of integro-differential equations. Both methods are based on a high-order Discontinous Galerkin approximation in space and the numerical inverse Laplace transform. In the first approach, we use functional calculus and the inverse Laplace transform to represent the solution. The spectral projections are then numerically computed and the approximation of the solution of the time-dependent problem is given by a summation of terms that are the product of projections of the data and the inverse Laplace transform of scalar functions. The second approach is the standard inverse Laplace transform technique. We show that the approach based on spectral projections can be very efficient when several time points are computed, and it is particularly interesting for parameter-dependent problems where the data or the kernel depends on a parameter.

Discontinuous Galerkin composite finite element methods (DGCFEM) are designed totackle approximation problems on complicated domains. Partial differential equationsposed on complicated domain are common when there are mesoscopic or local phenomena which need to be modelled at the same time as macroscopic phenomena. In thispaper, an optical lattice will be used to illustrate the performance of the approximationalgorithm for the ground state computation of a Gross–Pitaevskii equation, which isan eigenvalue problem with eigenvector nonlinearity. We will adapt the convergenceresults of Marcati and Maday 2018 to this particular class of discontinuous approximation spaces and benchmark the performance of the classic symmetric interior penaltyhp-adaptive algorithm against the performance of the hp-DGCFEM.

In this paper, we consider a sorting scheme for potentially spurious scattering resonant pairs in one- and two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can, therefore, be applied to structures with graded materials as well as to configurations including piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to mark potentially spurious solutions in electromagnetic and acoustic scattering resonances computations at a low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

In this paper, we study spectral properties and spectral enclosures for the Gurtin-Pipkin type of integro-differential equations in several dimensions. The analysis is based on an operator function and we consider the relation between the studied operator function and other formulations of the spectral problem. The theory is applied to wave equations with Boltzmann damping.

In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

In this paper, we study spectral properties and spectral enclosures for the Gurtin-Pipkin type of integro-differential equations in several dimensions. The analysis is based on an operator function and we consider the relation between the studied operator function and other formulations of the spectral problem. The theory is applied to wave equations with Boltzmann damping.