Consistency,convergence and stability of general discretizations of the initial value problem

Discretizations of nonlinear operators in Banach space are described and the concept of an inverse discretization introduced. In the main part of the paper, the very general formalism of BUTCHER for the initial value problem for ordinary differential equations is examined and the sufficiency of conditions for its stability and convergence is demonstrated. The order of convergence of these methods is discussed, and an example is given.     

On the stability and convergence of discretizations of initial value p.d.e.s

This paper examines the stability and convergence of discretizationsof initial value p.d.e.s using spatial discretization followedby time integration with an explicit one-step method. A Cauchyintegral representation is used to bound the growth in the discretesolution. New results are obtained regarding sufficient conditionsfor both algebraic and strong stability. Sufficient conditionsare also derived for convergence on a finite time interval.     

Automated FEM discretizations for the Stokes equation

Current FEM software projects have made significant advances in various automated modeling techniques. We present some of the mathematical abstractions employed by these projects that allow a user to switch between finite elements, linear solvers, mesh refinement and geometry, and weak forms with very few modifications to the code. To evaluate the modularity provided by one of these abstractions, namely switching finite elements, we provide a numerical study based upon the many different discretizations of the Stokes equations. AMS subject classification (2000)  74S05, 65Y99, 35Q30     

Time discretizations for evolution problems

The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed.     

Relaxations and discretizations for the pooling problem

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances.     

Innovative mimetic discretizations for electromagnetic problems

In this paper we introduce a discretization methodology for Maxwell equations based on Mimetic Finite Differences (MFD). Following the lines of the recent advances in MFD techniques (see Brezzi et al. (2007) [14] and the references therein) and using some of the results of Brezzi and Buffa (2007) [12], we propose mimetic discretizations for several formulations of electromagnetic problems both at low and high frequency in the time-harmonic regime. The numerical analysis for some of the proposed discretizations has already been developed, whereas for others the convergence study is an object of ongoing research.     

Regularising discretizations of the Birkhoff-Rott equation

A thin shear layer moving from the trailing edge of a two-dimensional aerofoil section downstream can be interpreted as a curve of discontinuity for the tangential velocity and may be approximated by a vortex sheet in inviscid, incompressible fluid flow. It is well known that vortex sheets are subject to instabilities of Kelvin-Helmholtz type which lead to roll-up phenomena in the wake. The motion of such sheets is governed by the Birkhoff-Rott equation. In the case of Kelvin-Helmholtz instability it seems clear that a curvature singularity occurs at a certain critical time and that consistent discretizations of the Birkhoff-Rott equation may fail to yield reliable results even before the time of occurrence of a singularity. We discuss the modification of the Biot-Savart kernel in the sense of Krasny who regularized the kernel by means of a global parameter. Using discrete Fourier transform we show the damping influence of this regularization technique. We modify the kernel carefully by introducing a regularization found in ordinary vortex methods and show that reliable results may be obtained up to and slightly after the singularity formation without increasing the accuracy of the computation.     

Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid

The authors construct the trajectory attractor and global attractor for an autonomous two-dimensional non-Newtonian fluid.     

Global attractor for the semilinear thermoelastic problem

In this paper we consider a class of semilinear thermoelastic problems. The global attractor for this semilinear thermoelastic problem with Dirichlet boundary condition is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Global attractor for Hirota equation

The long time behavior of solution for Hirota equation with zero order dissipation is studied. The global weak attractor for this system in Hper^k is constructed. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in Hper^k.     

Coding discretizations of continuous functions

We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical continuous function when the diameter of the discretization tends to zero. Our main result is that any finite word appears on a subsequence discretization with any desired limit frequency.     

A parallel solver for adaptive finite element discretizations

A parallel solver for the adaptive finite element analysis is presented. The primary aim of this work has been to establish an efficient parallel computational procedure which requires only local computations to update the solution of the system of equations arising from the finite element discretization after a local mesh-adaptation step. For this reason a set of algorithms has been developed (two-level domain decomposition, recursive hierarchical mesh-refinement, selective solution-update of linear systems of equations) which operate upon general and easily available partitioning, meshing and linear systems solving algorithms. AMS subject classification 15A23, 65N50, 65N60     

Exact eigensystems for some matrices arising from discretizations

It is known, for example, that the eigenvalues of the N×N matrix A, arising in the discretization of the wave equation, whose only nonzero entries are A_kk+1=A_k+1k=-1,k=1,…,N-1, and A_kk=2,k=1,…,N, are 2{1-cos[pπ/(N+1)]} with corresponding eigenvectors v^(p) given by . We show by considering a simple finite difference approximation to the second derivative and using the summation formulae for sines and cosines that these and other similar formulae arise in a simple and unified way.     

On additive Schwarz preconditioners for sparse grid discretizations

Summary Based on the framework of subspace splitting and the additive Schwarz scheme, we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems. For a BXP-like preconditioner we derive an estimate of the optimal orderO(1) and for a HB-like variant we obtain an estimate of the orderO(k ² ·2 ^k/2 ), wherek denotes the number of levels employed. Furthermore, we confirm these results by numerically computed condition numbers.     

Time discretizations for numerical optimisation of hyperbolic problems

We consider a class of numerical schemes for optimal control problems of hyperbolic conservation laws. We focus on finite-volume schemes using relaxation as a numerical approach to the optimality system. In particular, we study the arising numerical schemes for the adjoint equation and derive necessary conditions on the time integrator. We show that the resulting schemes are in particular asymptotic preserving for both, the adjoint and forward equation. We furthermore prove that higher-order time-integrator yields suitable Runge-Kutta schemes. The discussion includes the numerically interesting zero relaxation case.     

A note on flux limiting for diffusion discretizations

In this note, a limiting technique is presented to enforcemonotonicity for higher-order spatial diffusion discretizations.The aim is to avoid spurious oscillations and to improve thequalitative behaviour on coarse grids. The technique is relatedto known ones for convection equations, using limiters to boundthe numerical fluxes. Applications arise in pattern formationproblems for reaction–diffusion equations.