A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This paper demonstrates that these methods differ only in their choice of discrete inner product. Finally, certain uniqueness results for the covolume inner product are shown.

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +\nabla_g \cdotf(x,u)=0$ on a closed Riemannian manifold M.For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$

We consider a nonlinear second order elliptic boundary value problem (BVP)in a bounded domain $\Omega\subset {\mathbb R}^N$ witha nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant,accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part Γn . The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization algorithm are derived inL 2(Ω),H 1(Ω) andL ∞(Ω) spaces.

For scalar conservation laws in one space dimension with a flux function discontinuous inspace, there exist infinitely many classes of solutions which are L1 contractive.Each class is characterized by a connection (A,B) which determines the interface entropy. Forsolutions corresponding to a connection (A,B), there exists convergent numerical schemesbased on Godunov or Engquist−Osher schemes. The natural question is how to obtain schemes,corresponding to computationally less expensive monotone schemes like Lax−Friedrichs etc., usedwidely in applications. In this paper we completely answer this question for more general(A,B)stable monotone schemes using a novel construction of interface flux function. Then fromthe singular mapping technique of Temple and chain estimate of Adimurthi and Gowda, weprove the convergence of the schemes.

This paper is concerned with the dual formulation of the interface problemconsisting of a linear partial differential equation with variable coefficientsin some bounded Lipschitz domain Ω in $\mathbb{R}^n$ (n ≥ 2) and the Laplace equation with some radiation condition in theunbounded exterior domain Ωc := $\mathbb{R}^n\backslash\bar\Omega$ . The two problems are coupled by transmission andSignorini contact conditions on the interface Γ = ∂Ω. The exterior part of theinterface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequalitywith a linear operator. Then we treat the corresponding numerical scheme and discuss anapproximation of the NtD mapping with an appropriatediscretization of the inverse Poincaré-Steklov operator. In particular, assuming some abstract approximationproperties and a discrete inf-sup condition, we show unique solvability of the discrete scheme andobtain the corresponding a-priori error estimate. Next, we prove that these assumptions aresatisfied with Raviart-Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory.

We discuss a parallel implementation of the domaindecomposition method based on the macro-hybrid formulationof a second order ellipticequation and on an approximation by the mortar element method. The discretization leads to an algebraic saddle- point problem. An iterative method with a block- diagonalpreconditioner is used for solving the saddle- point problem. A parallel implementation of the method is emphasized. Finally the results of numerical experiments are presented.

We introduce a piecewise P2-nonconforming quadrilateralfinite element. First, we decompose a convex quadrilateral into the union of fourtriangles divided by its diagonals. Then the finite element space is defined by the set ofall piecewise P2-polynomials that are quadratic in eachtriangle and continuously differentiable on the quadrilateral. The degrees of freedom(DOFs) are defined by the eight values at the two Gauss points on each of the four edgesplus the value at the intersection of the diagonals. Due to the existence of one linearrelation among the above DOFs, it turns out the DOFs are eight. Global basis functions aredefined in three types: vertex-wise, edge-wise, and element-wise types. The correspondingdimensions are counted for both Dirichlet and Neumann types of elliptic problems. Forsecond-order elliptic problems and the Stokes problem, the local and global interpolationoperators are defined. Also error estimates of optimal order are given in both brokenenergy and L2(Ω) norms. The proposed elementis also suitable to solve Stokes equations. The element is applied to approximate eachcomponent of velocity fields while the discontinuousP1-nonconforming quadrilateral element is adopted toapproximate the pressure. An optimal error estimate in energy norm is derived. Numericalresults are shown to confirm the optimality of the presented piecewiseP2-nonconforming element on quadrilaterals.

We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1and 1 − ε, where ε > 0 is a small real parameter,i.e. the waveguide is gently converging. The width of the junction zonefor the semi-strips is 1 + O(√ε). We will present a sufficient condition for the existence of a weaklycoupled bound state below π2, the lower bound of thecontinuous spectrum. This eigenvalue in the discrete spectrum is unique and itsasymptotics is constructed and justified whenε → 0+.

Singularly perturbed reaction-diffusionproblems exhibit in general solutions with anisotropic features,e.g. strong boundary and/or interior layers.This anisotropy is reflected in a discretization by using mesheswith anisotropic elements. The quality of the numerical solutionrests on the robustness of the a posteriori error estimator withrespect to both, the perturbation parameters of the problemand the anisotropy of the mesh. The equilibrated residual method has been shown to provide one of the most reliable error estimates for the reaction-diffusion problem. Its modification suggested byAinsworth and Babuška has been proved to be robust for the case of singular perturbation. In the present work we investigate the modified method on anisotropic meshes. The method in the form of Ainsworth and Babuška is shown here to fail on anisotropic meshes. We suggest a new modification based on the stretching ratios of the mesh elements. The resulting error estimator is equivalent tothe equilibrated residual method in the case of isotropic meshesand is proved to be robust on anisotropic meshes as well. Among others, the equilibrated residual method involves the solution of an infinite dimensional local problem on each element. In practical computations an approximate solution to this local problem was successfully computed. Nevertheless, up to now no rigorous analysis has been done showing the appropriateness of any computable approximation. This demands special attention since an improper approximate solution to the local problem can be fatal for the robustness of the whole method. In the present work we provide one of the desired approximations. We prove that the method is not affected by the approximate solution of the local problem.

Cell-centered and vertex-centered finite volume schemes for the Laplace equationwith homogeneous Dirichlet boundary conditionsare considered on a triangular mesh and on the Voronoi diagram associated to its vertices.A broken P 1 function is constructed from the solutions of both schemes.When the domain is two-dimensional polygonal convex,it is shown that this reconstructionconverges with second-order accuracy towards the exact solution in the L 2 norm,under the sufficient condition that the right-hand side of the Laplace equation belongs to H 1(Ω).

We examine an elliptic optimal control problem with control and state constraints inℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisationinvolving piecewise constant functions for the control and piecewise linear for the state.The derived order of convergence is illustrated by a numerical example.

We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier [Lew and Buscaglia, Int. J. Numer. Methods Eng.76 (2008) 427–454]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional C 2-domains. For solution in Hq for q > 2, we prove that the method constructed with polynomials of degree one on each element approximates the function and its gradient with optimal orders h 2 and h, respectively. When q = 2, we have h 2-ε and h 1-ε for any ϵ > 0 instead. To this end,we construct a new interpolant that takes advantage of the discontinuities in the space, since standard interpolation estimates lead here to suboptimal approximation rates. The interpolation error estimate is based on proving an analog to Deny-Lions' lemma for discontinuous interpolants on a patch formed by the reference elements of any element and its three face-sharing neighbors. Consistency errors arising due to differences between the exact and the approximate domains are treated using Hardy's inequality together with more standard results on Sobolev functions.

In this paper we consider the Maxwell resolvent operator and its finite elementapproximation. In this framework it is natural the use of the edge elementspaces and to impose the divergence constraint in a weaksense with the introduction of a Lagrange multiplier, followingan idea by Kikuchi [14].We shall review some of the known properties for edge elementapproximations and prove some new result. In particular we shall prove auniform convergence in the L 2 norm for the sequence of discrete operators.These results, together with a general theory introduced by Brezzi, Rappaz andRaviart [8], allow an immediate proof of convergence for thefinite element approximation of the time-harmonicMaxwell system.

In this article we implement different numerical schemes to simulate the Schrödinger-Debye equations that occur in nonlinear optics. Since the existence of blow-up solutions is an open problem, we try to compute such solutions. The convergence of the methods is proved and simulations seem indeed to show that for at least small delays self-focusing solutions may exist.

In this short note we correct a conceptual error in theheuristic derivation of a kinetic equation used for thedescription of a one-dimensional granular medium in the socalled quasi-elastic limit, presented by the same authors inreference[1]. The equation we derived is however correct so that,the rigorous analysis on this equation, which constituted themain purpose of that paper, remains unchanged.

We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the cross sections appearing at the kinetic level.

Recently, adaptive wavelet strategies for symmetric, positive definite operators have been introduced that were proven to converge.This paper is devoted to the generalization to saddle point problems which are also symmetric, but indefinite. Firstly, we investigate a posteriori error estimates and generalize the known adaptive wavelet strategy to saddle point problems. The convergence of this strategy for elliptic operators essentially relies on the positive definite character of the operator. As an alternative, we introduce an adaptive variant of Uzawa's algorithm and prove its convergence. Secondly, we derive explicit criteria for adaptively refined wavelet spaces in order to fulfill the Ladyshenskaja-Babuška-Brezzi (LBB) condition and to be fully equilibrated.

We prove existence (uniqueness is easy) of a weak solution to a boundary value problem for an equation like $(v-1)^+_t = v_{xx} + F(v)_x$ where the function $F: \Bbb R\rightarrow\Bbb R$ is onlysupposed to be locally lipschitz continuous. In order to replace the lack of compactness in t on v<1, we use nonlinear semigroup theory.

Based on estimates for the KdV equation in analyticGevrey classes, a spectral collocation approximation ofthe KdV equation is proved to converge exponentially fast.

This article presents the derivation of a semi-classical model of electromagnetic-wave propagation in a non centro-symmetric crystal. It consists of Maxwell's equations for the wave field coupled with a version of Bloch's equations which takes fully into account the discrete symmetry group of the crystal. The model is specialized in the case of a KDP crystal for which information about the dipolar moments at the Bloch level can be recovered from the macroscopic dispersion properties of the material.