Reliable a-posteriori error estimators for hphp-adaptive finite element approximations of eigenvalue/eigenvector problems

We present reliable a-posteriori error estimates for $hp$-adaptive finite element approximations of eigenvalue/eigenvector problems. Starting from our earlier work on $h$ adaptive finite element approximations we show a way to obtain reliable and efficient a-posteriori estimates in the $hp$-setting. At the core of our analysis is the reduction of the problem on the analysis of the associated boundary value problem. We start from the analysis of Wohlmuth and Melenk and combine this with our a-posteriori estimation framework to obtain eigenvalue/eigenvector approximation bounds.Comment: submitte

Solving elliptic eigenvalue problems on polygonal meshes using discontinuous Galerkin composite finite element methods

In this paper we introduce a discontinuous Galerkin method on polygonal meshes. This method arises from the discontinuous Galerkin composite finite element method (DGFEM) for source problems on domains with micro-structures. In the context of the present paper, the flexibility of DGFEM is applied to handle polygonal meshes. We prove the a priori convergence of the method for both eigenvalues and eigenfunctions for elliptic eigenvalue problems. Numerical experiments highlighting the performance of the proposed method for problems with discontinuous coefficients and on convex and non-convex polygonal meshes are presented

Convergent adaptive finite element methods for photonic crystal applications

We prove the convergence of an adaptive finite element method for computing the band structure of 2D periodic photonic crystals with or without compact defects in both the TM and TE polarization cases. These eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficients. The error analysis extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to photonic crystal problems, and in particular deals with various complications which arise essentially from the lack of coercivity of the elliptic operator with discontinuous coefficients. We prove the convergence of the adaptive method in an oscillation-free way and with no extra assumptions on the initial mesh, beside the conformity and shape regularity. Also we present and prove the convergence of an adaptive method to compute efficiently an entire band in the spectrum. This method is guaranteed to converge to the correct global maximum and minimum of the band, which is a very useful piece of information in practice. Our numerical results cover both the cases of periodic structures with and without compact defects

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

In this paper we develop the a posteriori error estimation of hp-adaptive discontinuous Galerkin composite finite element methods (DGFEMs) for the discretization of second-order elliptic eigenvalue problems. DGFEMs allow for the approximation of problems posed on computational domains which may contain local geometric features. The dimension of the composite finite element space is independent of the number of geometric features. This is in contrast with standard finite element methods, as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space. Computable upper bounds on the error for both eigenvalues and eigenfunctions are derived. Numerical experiments highlighting the practical application of the proposed estimators within an automatic hp-adaptive refinement procedure will be presented

A hp-adaptive discontinuous Galerkin method for plasmonic waveguides

In this paper we propose and analyse a hphp-adaptive discontinuous finite element method for computing electromagnetic modes of propagation supported by waveguide structures comprised of a thin lossy metal film of finite width embedded in an infinite homogeneous dielectric. We propose a goal-oriented or dual weighted residual error estimator based on the solution of a dual problem that we use to drive the adaptive refinement with the aim to compute accurate approximation of the modes. We illustrate in the last section the benefits of the resulting hphp-adaptive method in practice, which consist in fast convergence and accurate estimation of the error. We tested the method computing the vanishing modes for a metallic waveguide of square section

Domain decomposition preconditioners for discontinuous Galerkin discretizations of compressible fluid flows

In this article we consider the application of Schwarz-type domain decomposition preconditioners to the discontinuous Galerkin finite element approximation of the compressible Navier-Stokes equations. To discretize this system of conservation laws, we exploit the (adjoint consistent) symmetric version of the interior penalty discontinuous Galerkin finite element method. To define the necessary coarse-level solver required for the definition of the proposed preconditioner, we exploit ideas from composite finite element methods, which allow for the definition of finite element schemes on general meshes consisting of polygonal (agglomerated) elements. The practical performance of the proposed preconditioner is demonstrated for a series of viscous test cases in both two- and three-dimensions

On Effects of Perforated Domains on Parameter-Dependent Free Vibration

Free vibration characteristics of thin perforated shells of revolution vary depending not only on the dimensionless thickness of the shell but also on the perforation structure. All holes are assumed to be free, that is, without any kinematical constraints. For a given conguration there exists a critical value of the dimensionless thickness below which homogenisation fails, since the modes do not have corresponding counterparts in the non-perforated reference shell. For a regular g g-perforation pattern, the critical thickness is reached when the lowest mode has an angular wave number of g=2. This observation is supported both by geometric arguments and numerical experiments. The numerical experiments have been carried out have been computed in 2D with high-order nite element method supporting Pitkaranta's mathematical shell model

River loads of freshwater and nutrients in the continental shelf area of the Northern Adriatic Sea

River discharges have a great effect on oceanographic properties and production processes in the marine ecosystem of the Northern Adriatic Sea. The combination between meteorological forcings and the spreading of river waters, mainly from the Po River, determine formation and dynamics of the coastal fronts in the Western and Northern part of this continental shelf area, through the generation of alternated vertical and horizontal density gradients that drive the circulation of surface waters and the pattern of Southward flowing West Adriatic Current. High inputs of river borne nutrients sustain the peaks of production in this marine system, in particular during spring and autumn. However, they may also cause hypoxic or anoxic crises in the deeper waters, in case of weak circulation. Strong density gradients coupled to alternate ambient conditions experienced by plankton communities in the coastal area, due to the dynamics of low and high salinity waters, are also at the basis of dystrophic events that have often occurred in this area, such as the appearance of large dinoflagellate blooms and mucilage phenomenon. Despite its basic importance, the role of river loads in the Northern Adriatic Sea has been poorly studied, to date. A number of studies were published in the literature, mostly for Po and Adige rivers, as a result of the efforts addressed to the mitigation of eutrophication problems in the coastal zones carried out during the 1970\u27 and the 1980\u27, while the comparison of river loads at sub-regional scale still remains largely incomplete. In the framework of VECTOR research project (sub-task 6.1.3. Compilation of river load data, loads of nutrients and dissolved and particulate organic matter), the analysis of the current importance of river discharges in the Northern Adriatic Sea has been carried out, as a part of the study of carbon biogeochemical cycle in this basin, taking in account monitoring data provided by several environmental agencies and scientific institutions of Italy, Slovenia and Croatia. From 2004 to 2007, water load by Po River (20.54 - 45.30 km3y-1) constituted, on the average, 66% of the total river load in the basin, Adige River (10%), Brenta River (7%) and Livenza River (6%) being the other important sources of freshwater. Despite Po is an highly significant proxy of the total river load in the Northern Adriatic, the inputs from the rivers located along the Northern and Eastern sections of the coast were not negligible (11 and 6% of the total, respectively) and often not in phase with the regime of Po River. As shown by distinct peaks of discharge and by prolonged drought periods that often occurred in the minor rivers differently from Po River, because of their more pronounced flashy flow regimes. During the same years, total nitrogen (86,000 - 262,000 t N y-1) and total phosphorus (3,840 - 9,500 t P y-1) loads of Po River were highly variable, mainly as a consequence of the oscillations of annual integrated water discharges. The transport of TN and TP was constituted by dissolved inorganic nitrogen for 69% and by reactive phosphorus for 47%, whereas a high load of reactive silicon was also estimated (64,400 - 137,500 t Si y-1). The rivers located in the Northern and Eastern areas of the coast contributed respectively for 8% and 4% to the total load of TN in the basin, but only for 4% and 1% to the total load of TP. This finding pointed out that the strong decreasing gradient from West to East of nutrient supply in the Northern Adriatic might be further exacerbated in case of a selective reduction of the flows of minor rivers, due to oncoming natural changes or to larger anthropogenic usage of the continental waters. River loads estimated in this study do not strongly differ from the available data published in the scientific literature during the last decades, but they showed that the ecosystem of the Northern Adriatic Sea may experience a strong reduction (≈ -50%) of the supply of land borne nutrients during dry years, like in 2005. Recurring years characterised by extremely low discharges could have a great impact on the biogeochemistry of the whole Northern Adriatic basin

Numerical solution of distributed-order time-fractional diffusion-wave equations using Laplace transforms

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