A coupled multivalued model for ice streams and its numerical simulation

^** Email: nati{at}dma.uvigo.es^*** Email: durany{at}dma.uvigo.es^**** Email: anaisabel.munoz{at}urjc.es^***** Email: emanuele.schiavi{at}urjc.es^****** Email: carlosv{at}udc.es This paper deals with the numerical solution of a non-linearmodel describing a free-boundary problem arising in modern glaciology.Considering a shallow, viscous ice sheet flow along a soft,deformable bed, a coupled non-linear system of differentialequations can be obtained. Particularly, an obstacle problemis then deduced and solved in the framework of its complementarityformulation. We present the numerical solution of the resultingmultivalued system modelling the ice sheet non-Newtonian dynamicsdriven by the underlying drainage system. Our numerical resultsshow the existence of fast ice streams when positive wave-likeinitial conditions are considered. The solutions are numericallycomputed with a decoupling iterative method and finite-elementtechnique. A duality algorithm and a projected Gauss–Seidelmethod are the alternatives used to cope with the resultingvariational inequality while the explicit treatment, Newtonmethod or a duality method are proposed to deal with the non-linearsource term. Finally, the numerical solutions are physicallyinterpreted and some comparisons among the numerical methodsare then discussed.     

An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations

^** Email: paul.houston{at}nottingham.ac.uk^*** Corresponding author. Email: ilaria.perugia{at}unipv.it^**** Email: schoetzau{at}math.ubc.ca We introduce a residual-based a posteriori error indicator fordiscontinuous Galerkin discretizations of H(curl; )-ellipticboundary value problems that arise in eddy current models. Weshow that the indicator is both reliable and efficient withrespect to the approximation error measured in terms of a naturalenergy norm. We validate the performance of the indicator withinan adaptive mesh refinement procedure and show its asymptoticexactness for a range of test problems.     

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.     

Self-similar solutions to the isothermal compressible Navier-Stokes equations

^** Email: guo_zhenhua{at}iapcm.ac.cn^*** Email: jiang{at}iapcm.ac.cn We investigate the self-similar solutions to the isothermalcompressible Navier–Stokes equations. The aim of thispaper is to show that there exist neither forward nor backwardself-similar solutions with finite total energy. This generalizesthe results for the incompressible case in Neas, J., Rika, M.& verák, V. (1996, On Leray's self-similar solutionsof the Navier-Stokes equations. Acta. Math., 176, 283–294),and is consistent with the (unproved) existence of regular solutionsglobally in time for the compressible Navier–Stokes equations.     

The cyclic Barzilai--Borwein method for unconstrained optimization

^** Email: dyh{at}lsec.cc.ac.cn^*** Email: hager{at}math.ufl.edu^**** Email: klaus.schittkowski{at}uni-bayreuth.de^***** Email: hzhang{at}math.ufl.edu In the cyclic Barzilai–Borwein (CBB) method, the sameBarzilai–Borwein (BB) stepsize is reused for m consecutiveiterations. It is proved that CBB is locally linearly convergentat a local minimizer with positive definite Hessian. Numericalevidence indicates that when m > n/2 3, where n is the problemdimension, CBB is locally superlinearly convergent. In the specialcase m = 3 and n = 2, it is proved that the convergence rateis no better than linear, in general. An implementation of theCBB method, called adaptive cyclic Barzilai–Borwein (ACBB),combines a non-monotone line search and an adaptive choice forthe cycle length m. In numerical experiments using the CUTErtest problem library, ACBB performs better than the existingBB gradient algorithm, while it is competitive with the well-knownPRP+ conjugate gradient algorithm.     

A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection-diffusion equations

^** Corresponding author. Email: l.elalaoui{at}imperial.ac.uk^*** Email: ern{at}cermics.enpc.fr^**** Email: erik.burman{at}epfl.ch We analyse a non-conforming finite-element method to approximateadvection–diffusion–reaction equations. The methodis stabilized by penalizing the jumps of the solution and thoseof its advective derivative across mesh interfaces. The a priorierror analysis leads to (quasi-)optimal estimates in the meshsize (sub-optimal by order in the L²-norm and optimal in thebroken graph norm for quasi-uniform meshes) keeping the Pécletnumber fixed. Then, we investigate a residual a posteriori errorestimator for the method. The estimator is semi-robust in thesense that it yields lower and upper bounds of the error whichdiffer by a factor equal at most to the square root of the Pécletnumber. Finally, to illustrate the theory we present numericalresults including adaptively generated meshes.     

The Helmholtz equation in a locally perturbed half-plane with passive boundary

^** Email: mduran{at}ing.puc.cl^*** Email: ignacio.muga{at}ucv.cl^**** Email: nedelec{at}cmapx.polytechnique.fr In this article, we study the existence and uniqueness of outgoingsolutions for the Helmholtz equation in locally perturbed half-planeswith passive boundary. We establish an explicit outgoing radiationcondition which is somewhat different from the usual Sommerfeld'sone due to the appearance of surface waves. We work with thehelp of Fourier analysis and a half-plane Green's function framework.This is an extended and detailed version of the previous articleDurán et al. (2005, The Helmholtz equation with impedancein a half-plane. C. R. Acad. Sci. Paris, Ser. I, 340, 483–488).     

On computing eigenvalues of second-order linear pencils

^** Email: mhannaby{at}yahoo.com^*** Email: zahraa26{at}yahoo.com In this paper, we use sinc techniques to compute the eigenvaluesof a second-order operator pencil of the form Q – P approximately.Here Q and P are self-adjoint differential operators of thesecond and first order, respectively. Also the eigenparameterappears in the boundary conditions linearly.     

Solving the continuous space p-centre problem: planning application issues

^** Email: wei.97{at}osu.edu^*** Email: murray.308{at}osu.edu^**** Email: xiao.37{at}osu.edu The Voronoi diagram heuristic has been proposed for solvingthe p-centre problem in continuous space. However, importantassumptions underlie this heuristic and may be problematic forpractical applications. These simplifying assumptions includeuniformly distributed demand, representing a region as a rectangle;analysis of a simple Voronoi polygon in solving associated one-centreproblems and no restrictions on potential facility locations.In this paper, we explore the complexity of solving the continuousspace p-centre problem in location planning. Considering theissue of solution space feasibility, we present a spatiallyrestricted version of this problem and propose methods for solvingit heuristically. Theoretical and empirical results are provided.     

The Z-transform and linear multistep stability

^** Email: mapjjc{at}maths.bath.ac.uk^*** Corresponding author. Email: ath{at}maths.bath.ac.uk^**** Email: hl{at}maths.bath.ac.uk This paper makes systematic use of control-theoretic methodssuch as the -transform, small-gain theorems and frequency-domainstability criteria in the analysis of the stability behaviourof linear multistep methods. Some of the results in Nevanlinna'swork are recovered and a number of new boundedness and asymptoticproperties of solutions of numerical schemes are obtained. Inparticular, we give a careful and detailed analysis of the nonlinearstability properties of strictly zero-stable methods.     

Post-processing with computable error bounds for the finite element approximation of a nonlinear heat conduction problem

Email: ain{at}mcs.le.ac.uk Email: D.Kelly{at}unsw.edu.au^* Email: I.Sloan{at}unsw.edu.au^** Email: swang{at}cs.curtin.edu.au It is shown how the finite element approximation of a nonlinearheat conduction problem may be post-processed to yield enhancedapproximations to the solution and the flux at any point inthe domain. Sharp computable bounds on the accuracy of the post-processedapproximations are derived. A criterion is identified for guidingadaptive refinements of the finite element discretization. Anumerical example is given illustrating the theoretical results.     

Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks

^** Email: belhach{at}poncelet.univ-metz.fr^*** Email: bucur{at}math.univ-metz.fr^**** Email: jmse{at}math.univ-metz.fr We study the Neumann–Laplacian eigenvalue problem in domainswith multiple cracks. We derive a mixed variational formulationwhich holds on the whole geometric domain (including the cracks)and implements efficient finite-element discretizations forthe computation of eigenvalues. Optimal error estimates aregiven and several numerical examples are presented, confirmingthe efficiency of the method. As applications, we numericallyinvestigate the behaviour of the low eigenvalues in domainswith a large number of cracks.     

A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension

^** Email: eymard{at}math.univ-mlv.fr^*** Email: gallouet{at}cmi.univ-mrs.fr^**** Corresponding author. Email: herbin{at}cmi.univ-mrs.fr Finite-volume methods for problems involving second-order operatorswith full diffusion matrix can be used thanks to the definitionof a discrete gradient for piecewise constant functions on unstructuredmeshes satisfying an orthogonality condition. This discretegradient is shown to satisfy a strong convergence property forthe interpolation of regular functions, and a weak one for functionsbounded in a discrete H¹-norm. To highlight the importance ofboth properties, the convergence of the finite-volume schemefor a homogeneous Dirichlet problem with full diffusion matrixis proven, and an error estimate is provided. Numerical testsshow the actual accuracy of the method.     

A stabilized finite-element method for the Stokes problem including element and edge residuals

^** Email: raraya{at}ing-mat.udec.cl^*** Corresponding author. Email: gbarrene{at}ing-mat.udec.cl^**** Email: valentin{at}lncc.br A new stabilized finite-element method is presented for theStokes problem. The method is of a Douglas–Wang type,and includes a positive jump term controlling the residual ofthe Cauchy stress tensor on the internal edges of the triangulation.A priori error estimates are obtained in the natural norms ofthe unknowns and an a posteriori error estimator is proposed,analysed and tested through numerical experiments.     

Geometry of Hugoniot curves in 2 x 2 systems of hyperbolic conservation laws with quadratic flux functions

^** Email: asakura{at}isc.osakac.ac.jp^*** Email: yamazaki{at}math.tsukuba.ac.jp This note analyzes a simple discontinuous solution to hyperbolic2 x 2 systems of conservation laws having quadratic flux functionswith an isolated umbilic point where the characteristic speedsare equal. We study the Hugoniot curves in Schaeffer & Shearer'scase I and II which are relevant to the three-phase Buckley–Leverettmodel for oil reservoir flow. The compressive and overcompressiveparts are determined. The wave curves through the umbilic pointare discussed and their compressive and overcompressive partsare also determined.     

Variable neighbourhood search for redundancy allocation problems

^** Email: ycliang{at}saturn.yzu.edu.tw^*** Email: s929512{at}mail.yzu.edu.tw^**** Email: s927522{at}mail.yzu.edu.tw A variable neighbourhood search (VNS) algorithm has been developedto solve the redundancy allocation problem (RAP). The VNS methodis perfectly suited to those combinatorial problems with potentialneighbourhood structures, as in the case of the RAP. The moststudied configuration of the RAP is a series system of s-independentk-out-of-n:G subsystems the so-called series–parallelsystem. The RAP is to select the optimal combination and redundancylevels of components to meet system-level constraints. Two typesof objectives are considered in this study—system reliabilitymaximization and system cost minimization. The VNS algorithmis tested on sets of benchmark problems and compared to thebest heuristics in the literature such as tabu search, multipleweighted objective heuristic, ant colony optimization and geneticalgorithm. Computational results show the advantages and benefitsof VNS for solving both types of RAP while considering bothsolution quality and computational efficiency.     

On the numerical solution of involutive ordinary differential systems: enhanced linear algebra

^** Email: jukka.tuomela{at}joensuu.fi^*** Corresponding author. Email: arponen{at}maths.warwick.ac.uk^**** Email: villesamuli.normi{at}joensuu.fi We analyse some Runge–Kutta type methods for computing1D integral manifolds, i.e. solutions to ordin-ary differentialequations and differential-algebraic equations. We show thatwe can compute the solutions which respect all the constraintsof the problem reliably and reasonably quickly. Moreover, weshow that the so-called impasse points are regular points inour approach and hence require no special attention.     

On the solvability for the mixed-type Lyapunov equation

^** Email: xsf{at}math.pku.edu.cn^*** Email: mscheng{at}math.pku.edu.cn In this paper, the linear matrix equation X = AXB^* + BXA^* +Q is considered, which is called the mixed-type Lyapunov equation.Some necessary and sufficient conditions for the existence ofa unique solution are presented. Since a Hermitian positivesemidefinite solution is important from the application pointof view, some sufficient conditions for the existence of a Hermitianpositive semidefinite solution are derived.     

Updating and downdating an upper trapezoidal sparse orthogonal factorization{dagger}

^** Email: santos{at}ctima.uma.es^*** Corresponding Author. Email: pablito{at}ctima.uma.es We describe how to update and downdate an upper trapezoidalsparse orthogonal factorization, namely the sparse QR factorizationof A_k^T, where A_k is a ‘tall and thin’ full columnrank matrix formed with a subset of the columns of a fixed matrixA. In order to do this, we have adapted Saunders' techniquesof the early 1970s for square matrices, to rectangular matrices(with fewer columns than rows) by using the static data structureof George and Heath of the early 1980s but allowing row downdatingon it. An implicitly determined column permutation allows usto dispense with computing a new ordering after each update/downdate;it fits well into the LINPACK downdating algorithm and ensuresthat the updated trapezoidal factor will remain sparse. We giveall the necessary formulae even if the orthogonal factor isnot available, and we comment on our implementation using thesparse toolbox of MATLAB 5.     

Supraconvergence of a finite difference scheme for solutions in Hs(0, L)

^** Email: silvia{at}mat.uc.pt^*** Email: ferreira{at}mat.uc.pt^**** Email: grigo{at}math.tu-berlin.de In this paper we study the convergence of a centred finite differencescheme on a non-uniform mesh for a 1D elliptic problem subjectto general boundary conditions. On a non-uniform mesh, the schemeis, in general, only first-order consistent. Nevertheless, weprove for s (1/2, 2] order O(h^s)-convergence of solution andgradient if the exact solution is in the Sobolev space H^1+s(0,L), i.e. the so-called supraconvergence of the method. It isshown that the scheme is equivalent to a fully discrete linearfinite-element method and the obtained convergence order isthen a superconvergence result for the gradient. Numerical examplesillustrate the performance of the method and support the convergenceresult.