Global superconvergence in combinations of Ritz-Galerkin-FEM for singularity problems

AbstractThis paper combines the piecewise bilinear elements with the singular functions to seek the corner singular solution of elliptic boundary value problems. The global superconvergence rates O(h2−δ) can be achieved by means of the techniques of Lin and Yan (The Construction and Analysis of High Efficient FEM, Hobei University Publishing, Hobei, 1996) for different coupling strategies, such as the nonconforming constraints, the penalty integrals, and the penalty plus hybrid integrals, where δ(>0) is an arbitrarily small number, and h is the maximal boundary length of quasiuniform rectangles −qij used. A little effort in computation is paid to conduct a posteriori interpolation of the numerical solutions, uh, only on the subregion used in finite element methods. This paper also explores an equivalence of superconvergence between this paper and Z.C. Li, Internat. J. Numer. Methods Eng. 39 (1996) 1839–1857 and J. Comput. Appl. Math. 81 (1997) 1–17

Convergence analysis of a multigrid algorithm for the acoustic single layer equation

We present and analyze a multigrid algorithm for the acoustic single layer equation in two dimensions. The boundary element formulation of the equation is based on piecewise constant test functions and we make use of a weak inner product in the multigrid scheme as proposed in \cite{BLP94}. A full error analysis of the algorithm is presented. We also conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the eigenfunctions for the Laplace single layer operator

Solving Elliptic Problems with Singular Sources Using Singularity Splitting Deep Ritz Method

In this work, we develop an efficient solver based on neural networks for secondorder elliptic equations with variable coefficients and a singular source. This class of problems covers general point sources, line sources, and the combination of point-line sources and has a broad range of practical applications. The proposed approach is based on decomposing the true solution into a singular part that is known analytically using the fundamental solution of the Laplace equation and a regular part that satisfies a suitable modified elliptic PDE with a smoother source and then solving for the regular part using the deep Ritz method. A path-following strategy is suggested to select the penalty parameter for enforcing the Dirichlet boundary condition. Extensive numerical experiments in two-and multi-dimensional spaces with point sources, line sources, or their combinations are presented to illustrate the efficiency of the proposed approach, and a comparative study with several existing approaches based on neural networks is also given, which shows clearly its competitiveness for the specific class of problems. In addition, we briefly discuss the error analysis of the approach

Nonlinear structural vibrations by the linear acceleration method

Numerical integration method for calculating dynamic response of nonlinear elastic structure

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

Model-order reduction techniques for the numerical solution of electromagnetic wave scattering problems

It is the aim of this work to contribute to the development of model-order reduction (MOR) techniques for the field of computational electromagnetics in relation to the electric field integral equation (EFIE) formulation. The ultimate goal is to enable a fast-sweep analysis. In a fast-sweep problem, some parameter on which the original problem depends is varying and the problem must be solved as the parameter changes over a desired parameter range. The complexity of the original model prohibits its direct use in simulation to compute the results at every required point. However, one can use MOR techniques to generate reduced-order models (ROMs), which can be rapidly solved to characterise the parameter-dependent behaviour of the system over the entire parameter range. This thesis focus is to implement robust, fast and accurate MOR techniques with strict error controls, for application with varying parameters, using the EFIE formulations. While these formulations result in matrices that are significantly smaller relative to differential equation-based formulations, the matrices resulting from discretising integral equations are very dense. Consequently, EFIEs pose a difficult proposition in the generation of low-order accurate reduced order models. The MOR techniques presented in this thesis are based on the theory of Krylov projections. They are widely accepted as being the most flexible and computationally efficient approaches in the generation of ROMs. There are three main contributions attributed to this work. ² The formulation of an approximate extension of the Arnoldi algorithm to produce a ROM for an inhomogeneous contrast-sweep and source-sweep analysis. ² Investigation of the application of the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) technique to problems in which the system matrix has a nonlinear parameter dependence for EFIE formulations. ² The development of a fast full-wave frequency sweep analysis using the WCAWE technique for materials with frequency-dependent dielectric properties