A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

Stability and consistency of kinetic upwinding for advection-diffusion equations

^** Email: matorril{at}ust.hk^*** Email: makxu{at}ust.hk Numerical methods based on kinetic models of fluid flows, likethe so-called BGK scheme, are becoming increasingly popularfor the solution of convection-dominated viscous fluid equationsin a finite-volume approach due to their accuracy and robustness.Based on kinetic-gas theory, the BGK scheme approxi-mately solvesthe BGK kinetic model of the Boltzmann equation at each cellinterface and obtains a numerical flux from integration of thedistribution function. This paper provides the first analyticalinvestigations of the BGK-scheme and its stability and consistencyapplied to a linear advection–diffusion equation. Thestructure of the method and its limiting cases are discussed.The stability results concern explicit time marching and demonstratethe upwinding ability of the kinetic method. Furthermore, itsstability domain is larger than that of common finite-volumemethods in the under-resolved case, i.e. where the grid Reynoldsnumber is large. In this regime, the BGK scheme is shown toallow the time step to be controlled from the advection alone.We show the existence of a third-order ‘super-convergence’on coarse grids independent of the initial condition. We alsoprove a limiting order for the local consist-ency error andshow the error of the BGK scheme to be asymptotically firstorder on very fine grids. However, in advection-dominated regimessuper-convergence is responsible for the high accuracy of themethod.     

Adaptive frame methods for elliptic operator equations: the steepest descent approach

Massimo Fornasier^¶ Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università "La Sapienza" in Roma, Via Antonio Scarpa, 16/B, I-00161 Roma, Italy Rob Stevenson^|| Department of Mathematics, Utrecht University, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands ^{This paper is concerned with the development of adaptive numericalmethods for elliptic operator equations. We are particularlyinterested in discretization schemes based on wavelet frames.We show that by using three basic subroutines an implementable,convergent scheme can be derived, which, moreover, has optimalcomputational complexity. The scheme is based on adaptive steepestdescent iterations. We illustrate our findings by numericalresults for the computation of solutions of the Poisson equationwith limited Sobolev smoothness on intervals in 1D and L-shapeddomains in 2D.}     

Sections de la puissance tensorielle du fibre tautologique sur le schema de Hilbert des points d'une surface

We compute the space of global sections for the tensor powerof the tautological bundle on the Hilbert scheme of points Hilbⁿ(X)of a complex smooth projective surface X.     

A stable semi-discrete central scheme for the two-dimensional incompressible Euler equations

^** Email: dlevy{at}math.stanford.edu We derive a second-order, semi-discrete central-upwind schemefor the incompressible 2D Euler equations in the vorticity formulation.The reconstructed velocity field preserves an exact discreteincompressibility relation. We state a local maximum principlefor a fully discrete version of the scheme and prove it usinga convexity argument. We then show how similar convexity argumentscan be used to prove that the scheme maps certain Orlicz spacesinto themselves. The consequences of this result on the convergenceof the scheme are discussed. Numerical simulations support theexpected properties of the scheme.     

A finite-volume scheme for dynamic reliability models

^** Email: christiane.cocozza{at}univ-mlv.fr^*** Email: robert.eymard{at}univ-mlv.fr^**** Email: sophie.mercier{at}univ-mlv.fr In a model arising in the dynamic reliability study of a system,the probability of the state of the system is completely describedby the Chapman–Kolmogorov equations, which are scalarlinear hyperbolic partial differential equations coupled bytheir right-hand side, the solution of which are probabilitymeasures. We propose in this paper a finite-volume scheme toapproximate these measures. We show, thanks to the proof ofthe tightness of the approximate solution, that the conservationof the probability mass leads to a compactness property. Theconvergence of the scheme is then obtained in the space of continuousfunctions with respect to the time variable, valued in the setof probability measures on [graphic: see PDF] . We finally show on a numerical example the accuracy and efficiencyof the approximation method.     

On the Resolution of the Heat Equation with Discontinuous Coefficients

We give in this work some results about the existence and uniqueness with optimal regularity for solutions of a parabolic equation in nondivergence form in L^q(0,T;L^p(Omega)) where 1 < p,q < infinity in two cases. We use Lamberton's results (cf. [9]) in the first case and Dore-Venni's results (cf. [6]) in the second case.     

A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov–Galerkinmethod with quadrature, for solving second-order, variable coefficient,elliptic boundary value problems on rectangular domains. Inour scheme, the trial space consists of C² splines of degreer 3, the test space consists of C⁰ splines of degree r –2, and we use composite (r – 1)-point Gauss quadrature.We show existence and uniqueness of the approximate solutionand establish optimal order error bounds in H², H¹ and L² norms.     

An unfitted finite-element method for elliptic and parabolic interface problems

Bhupen Deka Department of Mathematics, Assam University, Silchar-788011, India ^{A finite-element discretization, independent of the locationof the interface, is proposed and analysed for linear ellipticand parabolic interface problems. We establish error estimatesof optimal order in the H1-norm and almost optimal order inthe L2-norm for elliptic interface problems. An extension toparabolic interface problems is also discussed and an optimalerror estimate in the L2(0, T;H1())-norm and an almost optimalorder estimate in the L2(0, T;L2())-norm are derived for thespatially discrete scheme. A fully discrete scheme based onthe backward Euler method is analysed and an optimal order errorestimate in the L2(0, T;H1())-norm is derived. The interfacesare assumed to be of arbitrary shape and smooth for our purpose.}     

Leading infrared logarithms for the <Emphasis Type="Italic">σ</Emphasis>-model with fields on an arbitrary Riemann manifold

We derive a nonlinear recurrence equation for the infrared leading logarithms (LLs) in the four-dimensional σ-model with fields on an arbitrary Riemann manifold. The derived equation allows computing the LLs to an essentially unlimited loop order in terms of the geometric characteristics of the Riemann manifold. We reduce solving the SU(∞) principal chiral field in an arbitrary number of dimensions in the LL approximation to solving a very simple recurrence equation. This result prepares a way to solve the model in an arbitrary number of dimensions as N → ∞.     

On the Structure of the Moving Finite-element Equations

Address from 1st April 1985, School of Mathematics, Universityof Bristol, University Walk, Bristol BS8 1TW. The morning finite-element method for evolutionary partial differentialequations leads to a coupled non-linear system of ordinary differentialequations in time, with a coefficien matrix A, say, for thetime derivaties, We show for linear elements in any number ofdimensions, A can be written in the form M^TCM, where the matrixC depends solely on the mesh geometry and the matrix M on thegradient of the section, As a simple consequence we show thatA is singular only in the cases (i) element degeneracy () and (ii) collinearity of nodes (M not out of fullrank). We give constructions for the inversion of A in all cases. In one dimension, if A is non-singular, it has a simple explicitinverse. If A is singular we replace it by reduced matrix A^*.It can be shown that every case the spectral radius of the Jacobiiteration matrix ia ?and that A or A^* can be efficiently invertedby conjugate gradient methods. Finally, we discuss the applicability of these arguments tosystem of equations in any number of dimensions.     

An L2 finite element approximation for the incompressible Navier–Stokes equations

This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least-squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L²-approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second-order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L²-approximation to the stationary incompressible Navier–Stokes equations. The validity of L²-approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

A Fast Galerkin Scheme for Linear Integro-differential Equations

We describe an expansion method for the solution of first orderand second order ordinary integro-differential equations, whichis a generalization of the Fast Galerkin scheme for second kindintegral equations (Delves, 1977a; Delves, Abd-Elal & Hendry,1979). The method retains the O(N² In N) operation count ofthat scheme, and pays particular attention to the way in whichthe boundary conditions are incorporated, with the aim of retainingalso the stable structure of the Fast Galerkin equations, andits very rapid convergence. An error analysis, and numericalexamples, indicate that these aims are met.     

Error estimates for stochastic differential games: the adverse stopping case

^** Email: frederic.bonnans{at}inria.fr^*** Email: stefania.maroso{at}inria.fr^**** Email: zidani{at}ensta.fr We obtain error bounds for monotone approximation schemes ofa particular Isaacs equation. This is an extension of the theoryfor estimating errors for the Hamilton–Jacobi–Bellmanequation. To obtain the upper error bound, we consider the ‘Krylovregularization’ of the Isaacs equation to build an approximatesub-solution of the scheme. To get the lower error bound, weextend the method of Barles & Jakobsen (2005, SIAM J. Numer.Anal.) which consists in introducing a switching system whosesolutions are local super-solutions of the Isaacs equation.     

The fast multipole method for the symmetric boundary integral formulation

^** Email: of{at}mathematik.uni-stuttgart.de^*** Email: o.steinbach{at}tugraz.at^**** Email: wendland{at}mathematik.uni-stuttgart.de A symmetric Galerkin boundary-element method is used for thesolution of boundary-value problems with mixed boundary conditionsof Dirichlet and Neumann type. As a model problem we considerthe Laplace equation. When an iterative scheme is employed forsolving the resulting linear system, the discrete boundary integraloperators are realized by the fast multipole method. While thesingle-layer potential can be implemented straightforwardlyas in the original algorithm for particle simulation, the double-layerpotential and its adjoint operator are approximated by the applicationof normal derivatives to the multipole series for the kernelof the single-layer potential. The Galerkin discretization ofthe hypersingular integral operator is reduced to the single-layerpotential via integration by parts. We finally present a correspondingstability and error analysis for these approximations by thefast multipole method of the boundary integral operators. Itis shown that the use of the fast multipole method does notharm the optimal asymptotic convergence. The resulting linearsystem is solved by a GMRES scheme which is preconditioned bythe use of hierarchical strategies as already employed in thefast multipole method. Our numerical examples are in agreementwith the theoretical results.     

Generating Examples of Shock Propagation by Three Principles

In a medium characterized by a scalar speed C(x), a shock arrivesat the point x, after time T(x), with its magnitude decreasedby A(x). Symmetric C, T, and A in two dimensions can be convertedto cylindrically symmetric results in three dimensions by applyinga dimension-increasing principle: "Let C(x, y), T(x, y), andA(x, y) be even functions of y. They can be extended into threedimensions by using the formulas C(x, y)C(x, r), T(x, y)T(x,r), and A(x,y)A(x,r) [r^–1 cos(x, r)]^?, where r = (x²+2²)^?and is an auxiliary function." When C(x) is a function of asingle variable, the auxiliary function is given by cos(x,y) = T_y(x, y). In two dimensions, there is a conformal mappingprinciple: "Under the conformal mapping x+iy = f(x^*+iy^*), thefunctions T(x, y) and A(x, y) go into functions associated witha medium having speed C^*,y^*) = C(Re[f), Im[f]/f¹(x^*+iy."Thereis also an unchanged wavefronts principle: "If g is a smoothfunction with g(0) = 0 and g'(0)>0 then T^*(x) = g(T(x) andA^*(x) = A(x)[g'(x)/g'^1/2 are associated with a medium havingspeed C^*(x) = C(x)/g'(T(x))." in two dimensions, alternatingthe application of the last two principles generates a sequenceof media with their associated T(x, y) and A(x, y). Some ofthese can be extended into three dimensions by applying thefirst principle.     

On Milnor Fibrations of Arrangements

We use covering space theory and homology with local coefficientsto study the Milnor fiber of a homogeneous polynomial. Thesetechniques are applied in the context of hyperplane arrangements,yielding an explicit algorithm for computing the Betti numbersof the Milnor fiber of an arbitrary real central arrangementin C³, as well as the dimensions of the eigenspaces of the algebraicmonodromy. We also obtain combinatorial formulas for these invariantsof the Milnor fiber of a generic arrangement of arbitrary dimensionusing these methods.     

Convergence rates for the coupling of FEM and collocation BEM

We prove convergence of the coupling of finite and boundaryelements where Galerkin's methd is used for finite elementsand collocation for boundary elements. We consider linear ellipticboundary value problems in two dimensions, in particular problemsin elasticity. The mesh width k of the boundary elements andthe mesh width h of the finite elements are required to satisfykßh with suitable ß. Asymptotic error estimatesin the energy norm and in the L²-norm are derived. Numericalexamples are included.     

Cubic Splines on Curved Spaces

We consider a second-order problem in the calculus of variations,with an application to robotics in mind. The analysis is carriedout on a general Riemannian manifold M and then specializedto the case where M is the Lie group SO(3) of rotations in R³.For SO(3), the Euler-Lagrange equations reduce to interestingnonlinear systems of ordinary differential equations in R³.