Overlapping Schwarz methods for Maxwell''s equations in three dimensions

Summary. A two-level overlapping Schwarz method is considered for a Nédélec finite element approximation of 3D Maxwell's equations. For a fixed relative overlap, the condition number of the method is bounded, independently of the mesh size of the triangulation and the number of subregions. Our results are obtained with the assumption that the coarse triangulation is quasi-uniform and, for the Dirichlet problem, that the domain is convex. Our work generalizes well–known results for conforming finite elements for second order elliptic scalar equations. Numerical results for one and two-level algorithms are also presented. Received November 11, 1997 / Revised version received May 26, 1999 / Published online June 21, 2000     

Corrective meshless particle formulations for time domain Maxwell''s equations

In this paper a meshless approximation of electromagnetic (EM) field functions and relative differential operators based on particle formulation is proposed. The idea is to obtain numerical solutions for EM problems by passing up the mesh generation usually required to compute derivatives, and by employing a set of particles arbitrarily placed in the problem domain. The meshless Smoothed Particle Hydrodynamics method has been reformulated for solving the time domain Maxwell's curl equations. The consistency of the discretized model is investigated and improvements in the approximation are obtained by modifying the numerical process. Corrective algorithms preserving meshless consistency are presented and successfully used. Test problems, dealing with even and uneven particles distribution, are simulated to validate the proposed methodology, also by introducing a comparison with analytical solution.     

ADAPTIVITY IN SPACE AND TIME FOR MAGNETOQUASISTATICS

This paper addresses fully space-time adaptive magnetic field computations. We describe an adaptive Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell＇s equations on unstructured 3D tetrahedral grids. Spatial mesh re- finement and coarsening are based on hierarchical error estimators especially designed for combining tetrahedral H（curl）-conforming edge elements in space with linearly implicit Rosenbrock methods in time. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the magnetic recording write head benchmark problem proposed by the Storage Research Consortium in Japan.     

A high-order non-conforming discontinuous Galerkin method for time-domain electromagnetics

In this paper, we discuss the formulation, stability and validation of a high-order non-dissipative discontinuous Galerkin (DG) method for solving Maxwell’s equations on non-conforming simplex meshes. The proposed method combines a centered approximation for the numerical fluxes at inter element boundaries, with either a second-order or a fourth-order leap-frog time integration scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary-level hanging nodes. The method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities. Numerical experiments with high-order elements show the potential of the method.     

A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship, while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements, and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates. Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999     

Superconvergence analysis for Maxwell's equations in dispersive media

In this paper, we consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell's equations when dispersive media are involved.

     

Approximation dans un Problème de Viscoélasticité Périodique

We are concerned with equations which derive from a quasistatic periodic problem of viscoelasticity. We give a condition which yields to existence and uniqueness of a periodic solution. Then we prove a finite element method based on equilibrium elements for the space approximation and on the explicit Euler scheme for the time approximation.     

OPTIMAL ERROR ESTIMATES FOR NEDELEC EDGE ELEMENTS FOR TIME-HARMONIC MAXWELL'S EQUATIONS

In this paper, we obtain optimal error estimates in both L^2-norm and H（curl）-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell＇s equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L^2 error estimates into the L^2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec＇s second type elements, we present an optimal convergence estimate which improves the best results available in the literature.     

Internal and Boundary Observability Estimates for the Heterogeneous Maxwell's System

Observability estimates for Maxwell's system with variable coefficients are established using the differential geometry method recently developed for scalar wave equations. The main tool is that Maxwell's system is reducible to a perturbed vectorial wave equation with a decoupled principal part.     

Method of lines with boundary elements for 1‐D transient diffusion‐reaction problems

Time‐dependent differential equations can be solved using the concept of method of lines (MOL) together with the boundary element (BE) representation for the spatial linear part of the equation. The BE method alleviates the need for spatial discretization and casts the problem in an integral format. Hence errors associated with the numerical approximation of the spatial derivatives are totally eliminated. An element level local cubic approximation is used for the variable at each time step to facilitate the time marching and the nonlinear terms are represented in a semi‐implicit manner by a local linearization at each time step. The accuracy of the method has been illustrated on a number of test problems of engineering significance. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Convergence analysis of an adaptive edge element method for Maxwell's equations

An efficient and reliable a posteriori error estimate is derived for solving three-dimensional static Maxwell's equations by using the edge elements of first family. Based on the a posteriori error estimates, an adaptive finite element method is constructed and its convergence is established. Compared with the existing results, an important advantage of the new theory lies in its feature that the usual marking of elements based on the oscillation is not needed in our adaptive algorithm, while the linear convergence of the algorithm can be still demonstrated in terms of the reduction of the energy-norm error and the oscillation. Numerical examples are provided which demonstrate the effectiveness and robustness of the adaptive methods.     

A Nonconforming Arbitrary Quadrilateral Finite Element Method for Approximating Maxwell＇s Equations

The main aim of this paper is to provide convergence analysis of Quasi-Wilson nonconforming finite element to Maxwell's equations under arbitrary quadrilateral meshes. The error estimates are derived, which are the same as those for conforming elements under conventional regular meshes.     

A Nonconforming Arbitrary Quadrilateral Finite Element Method for Approximating Maxwell's Equations

The main aim of this paper is to provide convergence analysis of Quasi-Wilson nonconforming finite element to Maxwell's equations under arbitrary quadrilateral meshes.The error estimates are derived,which are the same as those for conforming elements under conventional regular meshes.     

Boundary integral equations for an interface linear crack under harmonic loading

The present study is devoted to application of boundary integral equations to the problem of a linear crack located on the bimaterial interface under time-harmonic loading. Using the Somigliana dynamic identity the system of boundary integral equations for displacements and tractions at the interface is derived. For the numerical solution the collocation method with piecewise constant approximation on each linear continuous boundary elements is used. The distributions of the displacements are computed for different values of the frequency of the incident tension-compression wave. Results are compared with static ones.     

A finite element method for approximating electromagnetic scattering from a conducting object

We provide an error analysis of a fully discrete finite element – Fourier series method for approximating Maxwell's equations. The problem is to approximate the electromagnetic field scattered by a bounded, inhomogeneous and anisotropic body. The method is to truncate the domain of the calculation using a series solution of the field away from this domain. We first prove a decomposition for the Poincaré-Steklov operator on this boundary into an isomorphism and a compact perturbation. This is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem. Using this decomposition, and edge elements to discretize the interior problem, we prove an optimal error estimate for the overall problem.     

Boundary element methods for Maxwell's equations on non-smooth domains

Summary. Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary element methods is proved. A sharp regularity result for the surface multipliers on polyhedral boundaries with plane faces is established. Received January 5, 2001 / Revised version received August 6, 2001 / Published online December 18, 2001 Correspondence to: C. Schwab     

A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations

This article considers a stabilized finite element approximation for the branch of nonsingular solutions of the stationary Navier–Stokes equations based on local polynomial pressure projection by using the lowest equal-order elements. The proposed stabilized method has a number of attractive computational properties. Firstly, it is free from stabilization parameters. Secondly, it only requires the simple and efficient calculation of Gauss integral residual terms. Thirdly, it can be implemented at the element level. The optimal error estimate is obtained by the standard finite element technique. Finally, comparison with other methods, through a series of numerical experiments, shows that this method has better stability and accuracy.     

A multiscale finite element method for the solution of transport equations

A multiscale Galerkin finite element scheme based on the residual free bubble function method is proposed to generate stable and accurate solutions for the transport equations namely diffusion-reaction (DR), convection-diffusion (CD) and convection-diffusion-reaction (CDR) equations. These equations show multiscale behavior in reaction or convection dominated situations. The idea is based on the approximation of the definite integral of the interpolation function within the element, instead of the function approximation. The numerical experiments are performed using the bilinear Lagrangian elements. To validate the approach, the numerical results obtained for a benchmark problem are compared with the analytical solution in a wide range of Peclet and Damköhler numbers. The results show that the developed method is capable of generating stable and accurate solutions.     

On the approximate solution of some two‐dimensional singular integral equations

An approximation method for a wide class of two‐dimensional integral equations is proposed. The method is based on using a special function system. Orthonormality and good interaction with fundamental integral operators arising in partial differential equations are remarkable properties of this system. In addition, all the basis elements can easily be calculated by recurrence relations. Taking into account these properties we construct a numerical algorithm which does not require additional effort (such as quadrature) to compute the values of the fundamental operators on the basis elements. Copyright © 2001 John Wiley & Sons, Ltd.     

拟线性对流占优扩散方程的扩展特征混合有限元方法数值模拟

讨论了拟线性对流占优扩散问题的数值模拟.对对流部分采用特征线格式进行离散,以消除流动锋线前沿的数值弥散现象,保证格式的稳定性;而对扩散部分采用扩展混合有限元方法,同时逼近未知函数,未知函数的梯度及伴随向量函数.理论分析和数值算例表明,此方法是稳定的,具有最优L2逼近精度.