Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains

Summary. In this paper, we study the spectral properties of Dirichlet problems for second order elliptic equation with rapidly oscillating coefficients in a perforated domain. The asymptotic expansions of eigenvalues and eigenfunctions for this kind of problem are obtained, and the multiscale finite element algorithms and numerical results are proposed. Mathematics Subject Classification (2000):65F10, 35P15This work is Supported by National Natural Science Foundation of China (grant # 19932030) and Special Funds for Major State Basic Research Projects (grant # TG2000067102)     

Algorithms for the matrix <Emphasis Type="Italic">p</Emphasis>th root

New theoretical results are presented about the principal matrix pth root. In particular, we show that the pth root is related to the matrix sign function and to the Wiener–Hopf factorization, and that it can be expressed as an integral over the unit circle. These results are used in the design and analysis of several new algorithms for the numerical computation of the pth root. We also analyze the convergence and numerical stability properties of Newtons method for the inverse pth root. Preliminary computational experiments are presented to compare the methods. AMS subject classification 15A24, 65H10, 65F30Numerical Analysis Report 454, Manchester Centre for Computational Mathematics, July 2004.Dario A. Bini: This work was supported by MIUR, grant number 2002014121.Nicholas J. Higham: This work was supported by Engineering and Physical Sciences Research Council grant GR/R22612 and by a Royal Society – Wolfson Research Merit Award.     

Well-posedness in <Emphasis Type="Italic">BV</Emphasis><Subscript><Emphasis Type="Italic">t</Emphasis></Subscript> and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units

Summary. We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We suggest to use a Krukov-type notion of entropy solution for this conservation law and prove uniqueness (L¹ stability) of the entropy solution in the BV_t class (functions W(x,t) with _tW being a finite measure). The existence of a BV_t entropy solution is established by proving convergence of a simple upwind finite difference scheme (of the Engquist-Osher type). A few numerical examples are also presented.Mathematics Subject Classification (2000):35L65, 35R05, 65M06, 76T20     

Adaptive polynomial preconditioning for hermitian indefinite linear systems

This paper explores the use of polynomial preconditioned CG methods for hermitian indefinite linear systems,Ax=b. Polynomial preconditioning is attractive for several reasons. First, it is well-suited to vector and/or parallel architectures. It is also easy to employ, requiring only matrix-vector multiplication and vector addition. To obtain an optimum polynomial preconditioner we solve a minimax approximation problem. The preconditioning polynomial,C(), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix,C(A)A. We also characterize the behavior of this minimax polynomial, which makes possible a thorough understanding of the associated CG methods. This characterization is also essential to the development of an adaptive procedure for dynamically determining the optimum polynomial preconditioner. Finally, we demonstrate the effectiveness of polynomial preconditioning in a variety of numerical experiments on a Cray X-MP/48. Our results suggest that high degree (20–50) polynomials are usually best.This research was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Dept. of Energy, by Lawrence Livermore National Laboratory under contract W-7405-ENG-48.This research was supported in part by the Dept. of Energy and the National Science Foundation under grant DMS 8704169.This research was supported in part by U.S. Dept. of Energy grant DEFG02-87ER25026 and by National Science Foundation grant DMS 8703226.     

Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis

Summary. The numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given Dirichlet data u_D by functions u_D,h in the trace space of a finite element space on _D. In this paper, quantitative a priori and a posteriori estimates are presented for two choices of u_D,h, namely the nodal interpolation and the orthogonal projection in L²(_D) onto the trace space. Two corresponding extension operators allow for an estimate of the boundary data approximation in global H¹ and L² a priori and a posteriori error estimates. The results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation.Mathematics Subject Classification (1991): 65N30, 65R20, 73C50This work was initiated while C. Carstensen was visiting the Max Planck Institute for Mathematics in the Sciences, Leipzig. S. Bartels acknowledges support by the German Research Foundation (DFG) within the Graduiertenkolleg Effiziente Algorithmen und Mehrskalenmethoden and the priority program Analysis, Modeling, and Simulation of Multiscale Problems. G. Dolzmann gratefully acknowledges partial support by the Max Planck Society and by the NSF through grant DMS0104118.     

Generalized Neuwirth Groups and Seifert Fibered Manifolds

The topological properties of the generalized Neuwirth groups, _n^k are discussed. For examp, we demonstrate that the group, _n^k is the fundamental group of the Seifert fibered space _n^k. Moreover, discuss some other invariants and algebraic properties of the above groups.This work was supported by Polish grant (BW-5100–5–0259–9) and the Russian Foundation for Basic Research (grant number 98–01–00699).2000 Mathematics Subject Classification: 20F34, 57M05, 57M60     

Total oscillation diminishing property for scalar conservation laws

Summary. We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone :, we have that |(u)|_TV() is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently in some numerical computations based on TVD schemes. We also show that semi-discrete Total Variation Diminishing finite volume schemes are TOD and that the fully discrete Godunov scheme is TOD.Mathematics Subject Classification (2000): 35L65, 35K55, 65M20     

Kinetic schemes for the relativistic gas dynamics

Summary. A kinetic solution for the relativistic Euler equations is presented. This solution describes the flow of a perfect gas in terms of the particle density n, the spatial part of the four-velocity u and the inverse temperature . In this paper we present a general framework for the kinetic scheme of relativistic Euler equations which covers the whole range from the non-relativistic limit to the ultra-relativistic limit. The main components of the kinetic scheme are described now. (i) There are periods of free flight of duration _M, where the gas particles move according to the free kinetic transport equation. (ii) At the maximization times t_n=n_M, the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by Jüttners relativistic generalization of the classical Maxwellian phase density. (iii) At each new maximization time t_n>0 we evaluate the so called continuity conditions, which guarantee that the kinetic scheme satisfies the conservation laws and the entropy inequality. These continuity conditions determine the new initial data at t_n. iv If in addition adiabatic boundary conditions are prescribed, we can incorporate a natural reflection method into the kinetic scheme in order to solve the initial and boundary value problem. In the limit _M0 we obtain the weak solutions of Eulers equations including arbitrary shock interactions. We also present a numerical shock reflection test which confirms the validity of our kinetic approach. Mathematics Subject Classification (1991):65M99, 76Y05This work is supported by the project Long-time behaviour of nonlinear hyperbolic systems of conservation laws and their numerical approximation, contract # DFG WA 633/7-2.     

Almost optimal interior penalty discontinuous approximations of symmetric elliptic problems on non-matching grids

Summary. We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on non-matching grids in this paper. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on partitioning of the domain into a finite number of subdomains. Further, an error analysis for the finite element approximation of the penalty formulation is given. Finally, numerical experiments on a series of model second order problems are presented. Mathematics Subject Classification (2000):65F10, 65N20, 65N30The work of the first and the second authors has been partially supported by the National Science Foundation under Grant DMS-9973328. The work of the last author was performed under the auspices of the U. S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract W-7405-Eng-48.     

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

Summary. We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation where generically for any given In addition to showing well-posedness of our approximation, we prove convergence in space dimensions $d \leq 3$. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented. Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 35K65, 35K35Supported by the EPSRC, U.K. through grant GR/M29689.Supported by the EPSRC, and by the DAAD through a Doktorandenstipendium     

Numerical Taylor expansions for invariant manifolds

Summary. We consider numerical computation of Taylor expansions of invariant manifolds around equilibria of maps and flows. These expansions are obtained by writing the corresponding functional equation in a number of points, setting up a nonlinear system of equations and solving this system using a simplified Newtons method. This approach will avoid symbolic or explicit numerical differentiation. The linear algebra issues of solving the resulting Sylvester equations are studied in detail.Mathematics Subject Classification (1991): 65Q05, 65P, 37M, 65P30, 65F20, 15A69Dedicated to Gerhard Wanner on the occasion of his 60th birthdayAcknowledgments. The authors like to thank Olavi Nevanlinna for discussions and his suggestion to use complex evaluation points.     

The excess of complex Hadamard matrices

A complex Hadamard matrix,C, of ordern has elements 1, –1,i, –i and satisfiesCC ^*=nI_nwhereC ^* denotes the conjugate transpose ofC. LetC=[c _ij] be a complex Hadamard matrix of order is called the sum ofC. (C)=|S(C)| is called the excess ofC. We study the excess of complex Hadamard matrices. As an application many real Hadamard matrices of large and maximal excess are obtained.Supported by an NSERC grant.Supported by Telecom grant 7027, an ATERB and ARC grant # A48830241.     

Convergence of flux vector splitting schemes with single entropy inequality for hyperbolic systems of conservation laws

Summary. We prove the convergence of flux vector splitting schemes associated to hyperbolic systems of conservation laws with a single compatible entropy _c. We prove estimate on the L² norm of the gradient of the numerical approximation in the inverse square root of the space increment x. This estimate is related to the notion of (strictly) _c-dissipativity on F⁺, –F^– and Id–(F⁺–F^–), where F⁺, F^– is the flux-decomposition. The second tool of the proof is a kinetic formulation of the flux-splitting scheme with three velocities. Then we get a control for all entropies and apply the compensated compactness theory.Mathematics Subject Classification (2000): 65M12, 35L65, 65M06, 82C40     

On circle containment orders

A partially ordered set is called acircle containment order provided one can assign to each element of the poset a circle in the plane so thatxy iff the circle assigned tox is contained in the circle assigned toy. It has been conjectured that every finite three-dimensional partially ordered set is a circle containment order. We show that the infinite three dimensional posetZ ³ isnot a circle containment order.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.Research supported in part by National Science Foundation, grant number DMS-8403646.     

Hearing pseudoconvexity with the Kohn Laplacian

A bounded domain in with connected Lipschitz boundary is pseudoconvex if the bottom of the essential spectrum of the Kohn Laplacian on the space of (0,q)-forms, 1qn–1, with L²-coefficients is positive.The author was supported in part by NSF grant DMS 0070697 and by an AMS centennial fellowship.Revised version: 9 July 2004     

Pointwise a posteriori error control for elliptic obstacle problems

We consider a finite element method for the elliptic obstacle problem over polyhedral domains in ^d, which enforces the unilateral constraint solely at the nodes. We derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle, which is just assumed to be Hölder continuous. They exhibit optimal order and localization to the non-contact set. We illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi-optimal meshes. Partially supported by NSF Grant DMS-9971450 and NSF/DAAD Grant INT-9910086.Partially suported by DAAD/NSF grant ``Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF'.Partially supported by DAAD/NSF grant ``Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF', and by the TMR network ``Viscosity solutions and their Applications', Italian M.I.U.R. projects ``Scientific Computing: Innovative Models and Numerical Methods' and ``Symmetries, Geometric Structures, Evolution and Memory in PDEs'.Mathematics Subject Classification (1991):65N15, 65N30, 35J85     

Rotations and Hermitian symmetric spaces

On an almost Hermitian manifold (M, g, J) one considers the naturally defined field of local diffeomorphismsj _m =exp _m J _m exp _m ^–1 ,mM, and in particular, one studies isometric, harmonic, holomorphic and symplecticj _m . This leads to some characterizations of special classes of almost Hermitian manifolds, including the class of Hermitian symmetric spaces. In addition, one treats some intrinsic and extrinsic geometrical properties of geodesic spheres relating to these local diffeomorphisms.Supported by grant 203.01.50 of the C.N.R., Italy.     

Additive Schwarz Methods for Elliptic Mortar Finite Element Problems

Summary. Two variants of the additive Schwarz method for solving linear systems arising from the mortar finite element discretization on nonmatching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed. The methods are defined on subdomains without overlap, and they use special coarse spaces, resulting in algorithms that are well suited for parallel computation. The condition number estimate for the preconditioned system in each method is proportional to the ratio H/h, where H and h are the mesh sizes, and it is independent of discontinuous jumps of the coefficients. For one of the methods presented the choice of the mortar (nonmortar) side is independent of the coefficients.This work has been supported in part by the Norwegian Research Council, grant 113492/420This work has been supported in part by the National Science Foundation, grant NSF-CCR-9732208 and in part by the Polish Science Foundation, grant 2P03A02116 Mathematics Subject Classification (2000):65N55     

Length functions of lemniscates

We study metric and analytic properties of generalized lemniscates E _t (f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E _t (f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E _t (f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulae of the length function in some special cases.The author was supported the Göran Gustafsson foundation and grant RFBR no. 03-01-00304.The author was supported by Russian President grant for young doctorates no. 00-15-99274 and grant RFBR no. 03-01-00304. Mathematics Subject Classification (2000):30E05, 42A82, 44A10     

A convergent monotone difference scheme for motion of level sets by mean curvature

Summary. An explicit convergent finite difference scheme for motion of level sets by mean curvature is presented. The scheme is defined on a cartesian grid, using neighbors arranged approximately in a circle. The accuracy of the scheme, which depends on the radius of the circle, dx, and on the angular resolution, d, is formally O(dx²+d). The scheme is explicit and nonlinear: the update involves computing the median of the values at the neighboring grid points. Numerical results suggest that despite the low accuracy, acceptable results are achieved for small stencil sizes. A numerical example is presented which shows that the centered difference scheme is non-convergent.Mathematics Subject Classification (2000): 35K65, 35K55, 65M06, 65M12The author would like to thank P.E. Souganidis for valuable discussions, and the University of Texas at Austin for its hospitality during the course of this work.