Energy conservation and super convergence analysis of the EC‐S‐FDTD schemes for Maxwell equations with periodic boundaries

This paper is concerned with new energy analysis of the two dimensional Maxwell's equations and the symmetric energy‐conserved splitting finite difference time domain (EC‐S‐FDTD) method with the periodic boundary (PB) condition. New energy identities of the Maxwell's equations in terms of H¹ and H² norms are proposed and interpreted by considering the physical meanings of the H¹ and H² semi‐norms in the identities. It is found from these new identities that the first and second curls of the electromagnetic fields are conserved in terms their magnitudes. By the energy methods, the numerical energy identities of the symmetric EC‐S‐FDTD method are derived and shown to converge to the continuous energy identities of the Maxwell's equations. This proves that the symmetric EC‐S‐FDTD scheme is unconditionally stable and energy conserved in the discrete H¹ and H² norms. Also by the energy methods, it is proved that the symmetric EC‐S‐FDTD method with PB condition is of second order (super) convergence in the discrete H¹ and H² norms. Numerical experiments are carried out and confirm the analysis on energy conservation, stability and super convergence.     

Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation

We use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L₂(0, T; H²(Ω)) ∩ L_∞(0, T; H¹(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C¹ cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.     

Approximation by herglotz wave functions

By a general argument, it is shown that Herglotz wave functions are dense (with respect to the C^∞(Ω)‐topology) in the space of all solutions to the reduced wave equation in Ω. This is used to provide corresponding approximation results in global spaces (eg. in L2‐Sobolev‐spaces H^m(Ω)) and for boundary data. Copyright © 2004 John Wiley & Sons, Ltd.     

A Two-dimensional Stationary Induction Heating Problem

We consider in this paper a system of equations modelling a steady-state induction heating process for ‘two-dimensional geometries’. Existence of a solution is stated in W^1,p(Ω) Sobolev spaces and is derived using the Leray–Schauder's fixed point theory. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Clifford algebras and Maxwell's equations in Lipschitz domains

We present a simple, Clifford algebra‐based approach to several key results in the theory of Maxwell's equations in non‐smooth subdomains of ℝ^m. Among other things, we give new proofs to the boundary energy estimates of Rellich type for Maxwell's equations in Lipschitz domains from [20, 10], discuss radiation conditions and the case of variable wave number. Copyright © 1999 John Wiley & Sons, Ltd.     

Ridged Domains,Embedding Theorems and Poincaré Inequalities

We study the embeddings E : W(X(Ω), Y(Ω)) ↪ Z(Ω), where X(Ω), Y(Ω) and Z(Ω) are rearrangement–invariant Banach function spaces (BFS) defined on a generalized ridged domain Ω, and W denotes a first–order Sobolev–type space. We obtain two–sided estimates for the measure of non–compactness of E when Z(Ω) = X(Ω) and, in turn, necessary and sufficient conditions for a Poincaré–type inequality to be valid and also for E to be compact. The results are used to analyse the example of a trumpet–shaped domain Ω in Lorentz spaces. We consider the problem of determining the range of possible target spaces Z(Ω), in which case we prove that the problem is equivalent to an analogue on the generalized ridge Γ of Ω. The range of target spaces Z(Ω) is determined amongst a scale of (weighted) Lebesgue spaces for “rooms and passages” and trumpet–shaped domains.     

Mixed finite element methods for generalized Forchheimer flow in porous media

Mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the flow of a single‐phase fluid in a porous medium in ?^d, d ≤ 3, subject to Forchhheimer's law—a nonlinear form of Darcy's law. Existence and uniqueness of the approximation are proved, and optimal order error estimates in L^∞(J; L²(Ω)) and in L^∞(J; H(div; Ω)) are demonstrated for the pressure and momentum, respectively. Error estimates are also derived in L^∞(J; L^∞(Ω)) for the pressure. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Existence for a time‐dependent heat equation with non‐local radiation terms

The paper deals with the time‐dependent linear heat equation with a non‐linear and non‐local boundary condition that arises when considering the radiation balance. Solutions are considered to be functions with values in V := {v ∈ H¹(Ω)∣γv ∈ L₅(∂Ω)}. As a consequence one has to work with non‐standard Sobolev spaces. The existence of solutions was proved by using a Galerkin‐based approximation scheme. Because of the non‐Hilbert character of the space V and the non‐local character of the boundary conditions, convergence of the Galerkin approximations is difficult to prove. The advantage of this approach is that we don't have to make assumptions about sub‐ and supersolutions. Finally, continuity of the solutions with respect to time is analysed. Copyright © 1999 John Wiley & Sons, Ltd.     

Regularity of weak solutions of Maxwell's equations with mixed boundary‐conditions

In this paper global H^s‐ and L^p‐regularity results for the stationary and transient Maxwell equations with mixed boundary conditions in a bounded spatial domain are proved. First it is shown that certain elements belonging to the fractional‐order domain of the Maxwell operator belong to H^s(Ω) for sufficiently small s > 0. It follows from this regularity result that H^s(Ω) is an invariant subspace of the unitary group corresponding to the homogeneous Maxwell equations with mixed boundary conditions. In the case that a possibly non‐linear conductivity is present a L^p‐regularity theorem for the transient equations is proved. Copyright © 1999 John Wiley & Sons, Ltd.     

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Discrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo and Omnès for the Laplace equation, are proposed for nonlinear diffusion problems in 2D with nonhomogeneous Dirichlet boundary condition. This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of nonconformal locally refined meshes. We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general W^?1,p′(Ω) source term and W^1‐(1/p),p(?Ω) boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient, respectively, in appropriate Lebesgue spaces. Finally, error estimates are given in the case where the solution is assumed to be in W^2,p(Ω). Numerical examples are given, including those on locally refined meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Regularity of the solutions of the steady‐state Boussinesq equations with thermocapillarity effects on the surface of the liquid

In this paper we show that every variational solution of the steady‐state Boussinesq equations ( u , p, θ) with thermocapillarity effect on the surface of the liquid has the following regularity: u ∈ H²(Ω)², p ∈ H¹(Ω), θ ∈ H²(Ω) under appropriate hypotheses on the angles of the ‘2‐D’ container (a cross‐section of the 3‐D container in fact) and of the horizontal surface of the liquid with the inner surface of the container. The difficulty comes from the boundary condition on the surface of the liquid (e.g. water) which modelizes the thermocapillarity effect on the surface of the liquid (equation (68.10) of Levich [7]). More precisely we will show that u ∈ P²₂(Ω)² and that θ ∈ P²₂(Ω), where P²₂(Ω) denotes the usual Kondratiev space. This result will be used in a forthcoming paper to prove convergence results for finite element methods intended to compute approximations of a non‐singular solution [1] of this problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Bivariate spline method for numerical solution of steady state Navier–Stokes equations over polygons in stream function formulation

We use the bivariate spline finite elements to numerically solve the steady state Navier–Stokes equations. The bivariate spline finite element space we use in this article is the space of splines of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the steady state Navier–Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth‐order equation, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in H²(Ω) of the nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The Galerkin method with C¹ cubic splines is implemented in MATLAB. Our numerical experiments show that the method is effective and efficient. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 147–183, 2000     

Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials

In this article, we consider the time‐dependent Maxwell's equations modeling wave propagation in metamaterials. One‐order higher global superclose results in the L² norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L^∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011     

Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem

We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.     

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H¹(Ω) × L₂(?Ω) with boundary data in L₂ due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H¹(Ω) into L₂(Ω), or L₂(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.     

Global well‐posedness and global attractor of fourth order semilinear parabolic equation

We study the initial boundary value problem of a class of fourth order semilinear parabolic equations. Global existence and nonexistence of solutions with initial data in the potential well are derived. Moreover, by using the iteration technique for regularity estimates, we obtain that for any k ≥ 0, the semilinear parabolic possesses a global attractor in H^k(Ω), which attracts any bounded subsets of H^k(Ω) in the H^k‐norm. Copyright © 2014 John Wiley & Sons, Ltd.     

Error estimates for the finite element approximation to a maxwell-type boundary value problem

Maxwell's boundary value problem for the time-harmonic case in a smooth, bounded domain G of R ² is considered. The optimal asymptotic L₂(G) and H₁(G)-error estimates 0(h²) and 0(h) resp, are derived for a piecewise linear finite element solution.     

An Inverse Function Theorem for Fréchet Spaces Satisfying a Smoothing Property and (DN)

Classical inverse function theorems of Nash-Moser type are proved for Fréchet spaces that admit smoothing operators as introduced by Nash. In this note an inverse function theorem is proved for Fréchet spaces which only have to satisfy the condition (DN) of Vogt and the smoothing property (S_Ω)_t; for instance, any Fréchet-Hilbert space which is an (Ω)-space in standard form has property (S_Ω)_t. The main result of this paper generalizes a theorem of Lojasiewicz and Zehnder. It can be applied to the space C^∞(K) if the compact K ? ?^N is the closure of its interior and subanalytic; different from classical results the boundary of K may have singularities like cusps. The growth assumptions on the mappings are formulated in terms of the weighted multiseminorms [ ]m,k introduced in this paper; nonlinear smooth partial differential operators on C^∞(K) and their derivatives satisfy these formal assumptions.     

Longtime behavior for a nonlinear wave equation arising in elasto‐plastic flow

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow u_tt?div{|?u|^m?1?u}?λΔu_t+Δ²u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X₁=H(Ω) × L²(Ω) and X=(H³(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd.     

Grand Sobolev spaces and their applications in geometric function theory and PDEs

Function spaces that are slightly larger than the Lebesgue L ^p (Ω) spaces (even larger than the Marcinkiewicz L ^p,∞ (Ω) spaces) have been introduced by Iwaniec and Sbordone [Arch. Ration. Mech. Anal. 119 (1992), 129–143] in connection with integrability properties of the Jacobian. These are the grand Lebesgue spaces L ^p)(Ω). In this survey we collect a number of results which prove that these spaces are useful in various classical settings of geometric function theory and partial differential equations (PDEs).