5 research outputs found
Numerical investigation of the conditioning for plane wave discontinuous Galerkin methods
We present a numerical study to investigate the conditioning of the plane
wave discontinuous Galerkin discretization of the Helmholtz problem. We provide
empirical evidence that the spectral condition number of the plane wave basis
on a single element depends algebraically on the mesh size and the wave number,
and exponentially on the number of plane wave directions; we also test its
dependence on the element shape. We show that the conditioning of the global
system can be improved by orthogonalization of the local basis functions with
the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES
iterations for solving the discrete problem iteratively.Comment: Submitted as a conference proceeding; minor revisio
Orbital-enriched Flat-top Partition of Unity Method for the Schr\"odinger Eigenproblem
Quantum mechanical calculations require the repeated solution of a
Schr\"odinger equation for the wavefunctions of the system. Recent work has
shown that enriched finite element methods significantly reduce the degrees of
freedom required to obtain accurate solutions. However, time to solution has
been adversely affected by the need to solve a generalized eigenvalue problem
and the ill-conditioning of associated systems matrices. In this work, we
address both issues by proposing a stable and efficient orbital-enriched
partition-of-unity method to solve the Schr\"odinger boundary-value problem in
a parallelepiped unit cell subject to Bloch-periodic boundary conditions. In
our proposed PUM, the three-dimensional domain is covered by overlapping
patches, with a compactly-supported, non-negative weight function, that is
identically equal to unity over some finite subset of its support associated
with each patch. This so-called flat-top property provides a pathway to devise
a stable approximation over the whole domain. On each patch, we use -th
degree orthogonal polynomials that ensure -th order completeness, and in
addition include eigenfunctions of the radial solution of the Schr\"odinger
equation. Furthermore, we adopt a variational lumping approach to construct a
block-diagonal overlap matrix that yields a standard eigenvalue problem and
demonstrate accuracy, stability and efficiency of the method.Comment: 24 pages, 12 figure
Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery
[EN] Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local error in quantities of interest (QoI), which are often more pertinent than local errors in energy for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems). These GOEE are one of the key unsolved problems of advanced engineering applications in, for example, the aerospace industry. This work presents a simple recovery-based error estimation technique for QoIs whose main characteristic is the use of an enhanced version of the Superconvergent Patch Recovery (SPR) technique previously used for error estimation in the energy norm. This enhanced SPR technique is used to recover both the primal and dual solutions. It provides a nearly statically admissible stress field that results in accurate estimations of the local contributions to the discretisation error in the QoI and, therefore, in an accurate estimation of this magnitude. This approach leads to a technique with a reasonable computational cost that could easily be implemented into already available finite element codes, or as an independent postprocessing tool.This work was supported by the EPSRC Grant EP/G042705/1 "Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method". Stephane Bordas also thanks partial funding for his time provided by the European Research Council Starting Independent Research Grant (ERC Stg Grant Agreement No. 279578) "RealTCut Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery". This work has received partial support from the research project DPI2010-20542 of the Ministerio de Economia y Competitividad (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politecnica de Valencia and Generalitat Valenciana (PROMETEO/2012/023) are also acknowledged. All authors also thank the partial support of the Framework Programme 7 Initial Training Network Funding under Grant No. 289361 "Integrating Numerical Simulation and Geometric Design Technology."González Estrada, OA.; Nadal Soriano, E.; RĂłdenas, J.; Kerfriden, P.; Bordas, S.; Fuenmayor Fernández, FJ. (2014). Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Computational Mechanics. 53(5):957-976. https://doi.org/10.1007/s00466-013-0942-8S957976535Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. 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