Time-discretization scheme for quasi-static Maxwell's equations with a non-linear boundary condition

We study a time dependent eddy current equation for the magnetic field H$\mathit{H}$ accompanied with a non-linear degenerate boundary condition (BC), which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electrical E$\mathit{E}$ and magnetic H$\mathit{H}$ fields obeys the following power law ν×E=ν×(|H×ν|^α-1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times (|\mathit{H}\times \mathit{\nu }{|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We establish the existence and uniqueness of a weak solution in a suitable function space under the minimal regularity assumptions on the boundary Γ$\Gamma$ and the initial data _H0${\mathit{H}}_0$. We design a non-linear time discrete approximation scheme based on Rothe's method and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time-discretization.     

Quasi-static Maxwell's equations with a dissipative non-linear boundary condition: Full discretization

We study a time dependent eddy current equation for the magnetic field H accompanied with a nonlinear boundary condition, which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electric (E) and magnetic (H  ) fields obeys the following power law (linearized for small and large values) ν×E=ν×(|H×ν|^α−1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times ({|\mathit{H}\times \mathit{\nu }|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We design a linear fully discrete approximation scheme to solve this nonlinear problem. The convergence of the approximations to a weak solution is proved, error estimates describing the dependence of the error on discretization parameters are derived as well. The efficiency of the proposed method is supported by numerical experiments.     

Convergence of the backward Euler method for type-II superconductors

Our paper is devoted to the study of a nonlinear degenerate transient eddy current problem of the type t_∂(|E|^−1/pE)+∇×(∇×E)=0, p>1, along with appropriate initial and boundary conditions. We design a nonlinear time-discrete numerical scheme for the approximation in suitable function spaces. We show the well-posedness of the problem, prove the convergence of the approximation to a weak solution and finally derive the error estimates. In the proofs, the monotonicity methods and the Minty-Browder argument are employed.     

Mixed finite element methods for general quadrilateral grids

We study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the H(div)-norm. This new element is designed so that the H(div)-projection Π_h satisfies ∇ · Π_h = P_hdiv. A rigorous optimal order error estimate is carried out by proving a modified version of the Bramble-Hilbert lemma for vector variables. We show that a local H(div)-projection reproducing certain polynomials suffices to yield an optimal L²-error estimate for the velocity and hence our approach also provides an improved error estimate for original Raviart-Thomas element of lowest order. Numerical experiments are presented to verify our theory.     

Nonlinear diffusion in type-II superconductors

In this paper we study a nonlinear evolution equation _∂t(σ(|E|)E)+∇×∇×E=F${\mathrm{\partial }}_t(\sigma (\left|\mathit{E}\right|)\mathit{E})+\nabla \times \nabla \times \mathit{E}=\mathit{F}$ in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field E$\mathit{E}$ in a conductive medium under the influence of a force F$\mathit{F}$. It is an approximation of Bean's critical-state model for type-II superconductors. We design a nonlinear numerical scheme for the time discretization. We prove the convergence of the proposed method. The proof is based on a generalization of div–curl lemma for transient problems. We also derive some error estimates for the approximate solution.     

MHD oblique stagnation-point flow of a micropolar fluid

The steady two-dimensional oblique stagnation-point flow of an electrically conducting micropolar fluid in the presence of a uniform external electromagnetic field (E₀, H₀) is analysed and some physical situations are examined. In particular, if E₀ vanishes, H₀ lies in the plane of the flow, with a direction not parallel to the boundary, and the induced magnetic field is neglected, it is proved that the oblique stagnation-point flow exists if, and only if, the external magnetic field is parallel to the dividing streamline. In all cases it is shown that the governing nonlinear partial differential equations admit similarity solutions and the resulting ordinary differential problems are solved numerically. Finally, the behaviour of the flow near the boundary is analysed; this depends on the three dimensionless material parameters, and also on the Hartmann number if H₀ is parallel to the dividing streamline.     

A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L ₂(T ^ν ×T ^ν ), where T ^ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrödinger operator ? = H₀ + H₁ + H₂, where H₀ is the operator of multiplication by a function and H₁ and H₂ are partial integral operators. We prove several theorems concerning the essential spectrum of ?. We study the discrete and essential spectra of the Hamiltonians H^t and h arising in the Hubbard model on the three-dimensional lattice.     

Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads

In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts.The numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), generalize the generalizations to coincidences, of Zhao's Nielsen number on the complement N(f;X−A), respectively Schirmer's relative Nielsen number N(f;X,A). Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps f^′ and k^′ which are homotopic as maps ofM-ads to f, respectively k firstly on the complement of the union of the subspaces Xν in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer's triad Nielsen number N(f;X₁∪X₂).Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here.     

Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H_∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.     

Exceptional sets related to Hayman's alternative

Let E be a subset of the complex plane C consisting of a countable set of points tending to ∞ and let k?1 be an integer. We derive a spacing condition (dependent on k) on the points of E which ensures that, if f is a function meromorphic in C with sufficiently large Nevanlinna deficiency at the poles, then either f takes every complex value infinitely often, or the kth derivative f(k) takes every non-zero complex value infinitely often, in C−E. This improves a previous result of Langley.     

A model of fracture for elliptic problems with flux and solution jumps

A new model of fracture for elliptic problems combining flux and solution jumps as immersed boundary conditions is proposed and proved to be well-posed. An application of this model to the flow in fractured porous media is also proposed including the cases of “impermeable fracture” and “fully permeable fracture” satisfying the so-called “cubic law”, as well as intermediate cases. A finite volume scheme on general polygonal meshes is built to solve such problems. Since no unknown is required at the fracture interface, the scheme is as cheap as standard schemes for the same problems without fault. The convergence of the scheme can be proved to the weak solution of the problem. With weak regularity assumptions, we also establish for the discrete H¹₀ and L² norms some error estimates in $\text{O}\text{(h)}$, where h is the maximum diameter of the control volumes of the mesh. To cite this article: Ph. Angot, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Finite element approximation of a free boundary plasma problem

In this article, we study a finite element approximation for a model free boundary plasma problem. Using a mixed approach (which resembles an optimal control problem with control constraints), we formulate a weak formulation and study the existence and uniqueness of a solution to the continuous model problem. Using the same setting, we formulate and analyze the discrete problem. We derive optimal order energy norm a priori error estimates proving the convergence of the method. Further, we derive a reliable and efficient a posteriori error estimator for the adaptive mesh refinement algorithm. Finally, we illustrate the theoretical results by some numerical examples.     

Equidistribution of Hecke Eigenforms on the Arithmetic Surface Γ0(N)\H

Given the orthonormal basis of Hecke eigenforms in S_2k(Γ(1)), Luo established an associated probability measure dμ_k on the modular surface Γ(1)\H that tends weakly to the invariant measure on Γ(1)\H. We generalize his result to the arithmetic surface Γ₀(N)\H where N?1 is square-free     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

Existence, regularity and local uniqueness of the solutions to the Maxwell-Landau-Lifshitz system in three dimensions

We study a model of ferromagnetism governed by Maxwell's equations coupled with the non-linear Landau-Lifshitz equation of micromagnetism. We are interested in the cases of space-periodic solutions for 3D domains. Obtaining the regularity of the solution m in space and of the solutions E, H in space we state the existence theorem. Finally, we prove the local uniqueness of the solutions.     

Extrapolated Crank-Nicolson approximation for a linear Stefan problem with a forcing term

In this paper, we apply finite element Galerkin method to a single-phase linear Stefan problem with a forcing term. We apply the extrapolated Crank-Nicolson method to construct the fully discrete approximation and we derive optimal error estimates in the temporal direction inL ²,H ¹ spaces.     

The hyperring of adèle classes

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We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK=AK/K^× of a global field K. After promoting F₁ to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R^× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G∪{0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3LSKD_PfJyc.     

Bounds for the error of linear systems of equations using the theory of moments

Consider the system, of linear equations Ax = b where A is an n × n real symmetric, positive definite matrix and b is a known vector. Suppose we are given an approximation to x, ξ, and we wish to determine upper and lower bounds for ∥ x − ξ ∥ where ∥ ··· ∥ indicates the euclidean norm. Given the sequence of vectors {r_i}_{i^k = 0}, where r_i = Ar_{i − 1} and r₀ = b − Aξ, it is shown how to construct a sequence of upper and lower bounds for ∥ x − ξ ∥ using the theory of moments.