On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

On the numerical solution of Plateau's problem

The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization γ of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization κ of the unit circle. The Dirichlet energy of the harmonic extension of γ○κ has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.     

Improved Convergence Results for the Finite Element Method with Holomorphic Functions

The purpose of this paper is to present extended results on convergence of a method for coupling of an analytical and a finite element solution for a boundary value problem with a singularity. As an example of a singular problem we consider a solution to the Lamé-Navier equation with a singularity caused by a crack. The main idea of such an approach is to construct a continuous coupling between an analytical solution near a singularity and a finite element solution through the whole interaction interface. In this paper we present a basic theory of convergence for a method of coupling, particularly we discuss some points which show a difference to the standard theory of the finite element method.     

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

Interface problem in holonomic elastoplasticity

The three-dimensional interface problem with the homogeneous Lamé system in an unbounded exterior domain and holonomic material behaviour in a bounded interior Lipschitz domain is considered. Existence and uniqueness of solutions of the interface problem are obtained rewriting the exterior problem in terms of boundary integral operators following the symmetric coupling procedure. The numerical approximation of the solutions consists in coupling of the boundary element method (BEM) and the finite element method (FEM). A Céa-like error estimate is presented for the discrete solutions of the numerical procedure proving its convergence.     

The Dual Singular Function Method for 2D Boundary Integral Equations

The accuracy of standard boundary element methods for elliptic boundary value problems deteriorates if the boundary of the domain contains corners or if the boundary conditions change along the boundary. Here we first investigate the convergence behaviour of standard spline Galerkin approximation on quasi-uniform meshes for boundary integral equations on polygonal domains. It turns out, that the order of convergence depends on some constant describing the singular behaviour of solutions near corner points of the boundary. In order to recover the full order of convergence for the Galerkin approximation we propose the dual singular function method which is often used for improving the accuracy of finite element methods. The theoretical convergence results are confirmed and illustrated by a numerical example.     

A note on a finite element method dealing with corner singularities

Recently the first author and his coworker report a new finite element method for the Poisson equations with homogeneous Dirichlet boundary conditions on a polygonal domain with one re-entrant angle [7]. They use the well-known fact that the solution of such problem has a singular representation, deduce a well-posed new variational problem for regular part of solution and an extraction formula for the so-called stress intensity factor using two cut-off functions. They use Fredholm alternative and G?rding’s inequality to establish the well-posedness of the variational problem and finite element approximation, so there is a maximum bound for meshh theoretically, although the numerical experiments shows the convergence for every reasonable size ofh. In this paper we show that the method converges for everyh with reasonable size by imposing a restriction to the support of the extra cut-off function without using G?rding’s inequality. We also give error analysis with similar results.     

Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data

This paper is concerned with solving the Cauchy problem for an elliptic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to adapted stopping criteria for the minimization process depending on the noise rate. Numerical examples involving smooth and singular data are presented.     

An hp finite element method for singularly perturbed systems of reactiondiffusion equations

We consider the approximation of a coupled system of two singularly perturbed reaction-diffusion equations by the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. We present results on a high order hp finite element scheme which includes elements of size O (εp) and O (μp) near the boundary, where ε, μ are the singular perturbation parameters and p is the degree of the approximating polynomials. Under the assumption of analytic input data, the method yields exponential rates of convergence as p → ∞, independently of ε and μ. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal convergence rates for the combined methods of different finite element methods

Coupling techniques are essential to combining different numerical methods together for the purpose of solving an elliptic boundary value problem. By means of nonconforming constraints, the combinations of various Lagrange finite element methods often cause reduced rates of convergence. In this article, we present a method using penalty plus hybrid technique to match different finite element methods such that the optimal convergence rates in the ‖ · ‖_h and zero norms of errors of the solution can always be achieved. Also, such a coupling technique will lead to an optimal asymptotic condition number for the associated coefficient matrix. Moreover, this study can easily be extended for combining the finite difference method with the finite element method to also yield the optimal rate of convergence.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

The discrete Plateau Problem: Algorithm and numerics

We solve the problem of finding and justifying an optimal fully discrete finite element procedure for approximating minimal, including unstable, surfaces. In this paper we introduce the general framework and some preliminary estimates, develop the algorithm, and give the numerical results. In a subsequent paper we prove the convergence estimate. The algorithmic procedure is to find stationary points for the Dirichlet energy within the class of discrete harmonic maps from the discrete unit disc such that the boundary nodes are constrained to lie on a prescribed boundary curve. An integral normalisation condition is imposed, corresponding to the usual three point condition. Optimal convergence results are demonstrated numerically and theoretically for nondegenerate minimal surfaces, and the necessity for nondegeneracy is shown numerically.

     

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

We study a second order hyperbolic initial‐boundary value partial differential equation (PDE) with memory that results in an integro‐differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 548–563, 2016     

High-order dual-parametric finite element methods for cavitation computation in nonlinear elasticity

In this paper, we present the numerical analysis on high order dual parametric finite element methods for the cavitation computation problems in nonlinear elasticity, which leads to a meshing strategy assuring high efficiency on numerical approximations to cavity deformations. Furthermore, to cope with the high order approximation of the finite element methods, properly chosen weighted Gaussian type numerical quadrature is applied to the singular part of the elastic energy. Our numerical experiments show that the high order dual parametric finite element methods work well when coupled with properly designed weighted Gaussian type numerical quadratures for the singular part of the elastic energy, and the convergence rates of the numerical cavity solutions are shown to be significantly improved as expected.     

NON C~0 NONCONFORMING ELEMENTS FOR ELLIPTIC FOURTH ORDER SINGULAR PERTURBATION PROBLEM

In this paper we give a convergence theorem for non C^0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem. Two such kind of elements, a nine parameter triangular element and a twelve parameter rectangular element both with double set parameters, are presented. The convergence and numerical results of the two elements are given.     

On the supports of functions

In this paper, finite difference and finite element methods are used with nonlinear SOR to solve the problems of minimizing strict convex functionals. The functionals are discretized by both methods and some numerical quadrature formula. The convergence of such discretization is guaranteed and will be discussed. As for the convergence of the iterative process, it is necessary to vary the relaxation parameter in each iterations. In addition, for the model catenoid problem, boundary grid refinements play an essential role in the proposed nonlinear SOR algorithm. Numerical results which illustrate the importance of the grid refinements will be presented.

     

Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems

In this article, we consider rectangular finite element methods for fourth order elliptic singular perturbation problems. We show that the non‐ C⁰ rectangular Morley element is uniformly convergent in the energy norm with respect to the perturbation parameter. We also propose a C⁰ extended high order rectangular Morley element and prove the uniform convergence. Finally, we do some numerical experiments to confirm the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the mixed finite element method with Lagrange multipliers

In this note we analyze a modified mixed finite element method for second‐order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R ². The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu?ka‐Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart‐Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192–210, 2003     

Numerical modelling of semi-coercive beam problem with unilateral elastic subsoil of Winkler’s type

A non-linear semi-coercive beam problem is solved in this article. Suitable numerical methods are presented and their uniform convergence properties with respect to the finite element discretization parameter are proved here. The methods are based on the minimization of the total energy functional, where the descent directions of the functional are searched by solving the linear problems with a beam on bilateral elastic “springs”. The influence of external loads on the convergence properties is also investigated. The effectiveness of the algorithms is illustrated on numerical examples.     

Superconvergence of Solution Derivatives of the Shortley–Weller Difference Approximation to Elliptic Equations with Singularities Involving the Mixed Type of Boundary Conditions

This paper presents a superconvergence analysis for the Shortley–Weller finite difference approximation of second-order self-adjoint elliptic equations with unbounded derivatives on a polygonal domain with the mixed type of boundary conditions. In this analysis, we first formulate the method as a special finite element/volume method. We then analyze the convergence of the method in a finite element framework. An O(h ^1.5)-order superconvergence of the solution derivatives in a discrete H ¹ norm is obtained. Finally, numerical experiments are provided to support the theoretical convergence rate obtained.