Keller's Box-Scheme for the One-Dimensional Stationary Convection-Diffusion Equation

u ,∇u)=f, is to take the average onto the same mesh of the two equations of the mixed form, the conservation law div p=f and the constitutive law p=ϕ(u,∇u). In this paper, we perform the numerical analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion equation cu _x −ɛu _xx =f. In the first one, introduced by B. Courbet in [9,10], the numerical average of the diffusive flux is upwinded along the sign of the velocity, giving a first order accurate scheme. The second one is fourth order accurate. It is based onto the Euler-MacLaurin quadrature formula for the average of the diffusive flux. We emphasize in each case the link with the SUPG finite element method. Received June 7, 2001; revised October 2, 2001     

H 1-Galerkin Mixed Finite Element Method for the Regularized Long Wave Equation

In this paper, an H¹-Galerkin mixed finite element method is proposed for the 1-D regularized long wave (RLW) equation u_t+u_x+uu_x−δu_xxt=0. The existence of unique solutions of the semi-discrete and fully discrete H¹-Galerkin mixed finite element methods is proved, and optimal error estimates are established. Our method can simultaneously approximate the scalar unknown and the vector flux effectively, without requiring the LBB consistency condition. Finally, some numerical results are provided to illustrate the efficacy of our method.     

A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L ² norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H ¹ norm. This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.     

Modified nonlinearly dispersive mK(m,n,k) equations: I. New compacton solutions and solitary pattern solutions

In this paper exact solutions of the modified nonlinearly dispersive KdV equations (simply called mK(m,n,k) equations), u^m−1u_t+a(uⁿ)_x+b(u^k)_xxx=0, are investigated by using some direct ansatze. As a result, abundant new compacton solutions: solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave solutions and periodic wave solutions are obtained.     

An Adaptive Time Step Procedure for a Parabolic Problem with Blow-up

In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blow-up of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme. Received October 4, 2001; revised March 27, 2002 Published online: July 8, 2002     

New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations

The special exact solutions of nonlinearly dispersive Boussinesq equations (called B(m,n) equations), u_tt−u_xx−a(uⁿ)_xx+b(u^m)_xxxx=0, is investigated by using four direct ansatze. As a result, abundant new compactons: solitons with the absence of infinite wings, solitary patterns solutions having infinite slopes or cups, solitary waves and singular periodic wave solutions of these two equations are obtained. The variant is extended to include linear dispersion to support compactons and solitary patterns in the linearly dispersive Boussinesq equations with m=1. Moreover, another new compacton solution of the special case, B(2,2) equation, is also found.     

Parabolic equations with double variable nonlinearities

The paper is devoted to the study of the homogeneous Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions:

ut=div(a(x,t,u)|u^|α(x,t)|∇u^|p(x,t)−2∇u)+f(x,t)     

Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q₁–P₀ quadrilateral element and the P₁–P₀ triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H²) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|^1/2H³). The methods we study provide an approximate solution (u^h,p^h) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.     

Discrete maximum principle for parabolic problems solved by prismatic finite elements

In this paper we analyze the discrete maximum principle (DMP) for a non-stationary diffusion–reaction problem solved by means of prismatic finite elements and θ$\theta$-method. We derive geometric conditions on the shape parameters of prismatic partitions and time-steps which a priori guarantee validity of the DMP. The presented numerical tests illustrate the sharpness of the obtained conditions.     

Error Estimates of a FEM with Lumping for Parabolic PDEs

L ²-norm. This well-known FEM is given by the use of the vertical line method and conforming linear finite elements on a triangulation. The main result of the paper are new estimates in the L ²-norm for the additional error term originated by lumping. Using these ones, for the FEM with lumping we can apply directly the proof technique of error estimates known for conforming FEMs. Received May 17, 2001; revised November 2, 2001 Published online February 18, 2002     

Accurate evaluation of divided differences for polynomial interpolation of exponential propagators

In this paper, we propose an approach to the computation of more accurate divided differences for the interpolation in the Newton form of the matrix exponential propagator φ(hA)v, φ (z) = (e ^z − 1)/z. In this way, it is possible to approximate φ (hA)v with larger time step size h than with traditionally computed divided differences, as confirmed by numerical examples. The technique can be also extended to “higher” order φ _k functions, k≥0.     

Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation Δ² u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v _h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in u _h , which is an approximation of u. A direct approximation v _h of v can be obtained from the approximation u _h of u. Using piecewise polynomials of degree p≥3, a priori error estimates of u−u _h in the broken H ¹ norm as well as in L ² norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v−v _h in L ² norm which is suboptimal in h and p is also discussed. When p=2, the preset method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems

In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.     

Finite Volume Element Approximations of Nonlocal in Time One-Dimensional Flows in Porous Media

Various finite volume element schemes for parabolic integro-differential equations in 1-D are derived and studied. These types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. We consider the lowest-order (linear and L-splines) finite volume elements, although higher-order volume elements can be considered as well under this framework. It is proved that finite volume element approximations are convergent with optimal order in H ¹-norms, suboptimal order in the L ²-norm and super-convergent order in a discrete H ¹-norm. Received August 3, 1998; revised October 11, 1999     

Some new paranormed sequence spaces

Maddox defined the sequence spaces ℓ_∞(p), c(p) and c₀(p) in [Proc. Camb. Philos. Soc. 64 (1968) 335, Quart. J. Math. Oxford (2) 18 (1967) 345]. In the present paper, the sequence spaces a₀^r(u,p) and a_c^r(u,p) of non-absolute type have been introduced and proved that the spaces a₀^r(u,p) and a_c^r(u,p) are linearly isomorphic to the spaces c₀(p) and c(p), respectively. Besides this, the α-, β- and γ-duals of the spaces a₀^r(u,p) and a_c^r(u,p) have been computed and their basis have been constructed. Finally, a basic theorem has been given and later some matrix mappings from a₀^r(u,p) to the some sequence spaces of Maddox and to some new sequence spaces have been characterized.     

The Fourier-Nitsche-mortaring for elliptic problems with reentrant edges

The Fourier method is combined with the Nitsche-finite-element method (as a mortar method) and applied to the Dirichlet problem of the Poisson equation in three-dimensional axisymmetric domains with reentrant edges generating singularities. The approximating Fourier method yields a splitting of the 3D problem into a set of 2D problems on the meridian plane of the given domain. For solving the 2D problems bearing corner singularities, the Nitsche-finite-element method with non-matching meshes and mesh grading near reentrant corners is applied. Using the explicit representation of some singularity function of non-tensor product type, the rate of convergence of the Fourier-Nitsche-mortaring is estimated in some H ¹-like norm as well as in the L ₂-norm for weak regularity of the solution. Finally, some numerical results are presented.      

Computing multiple pitchfork bifurcation points

A point (x*,λ*) is called apitchfork bifurcation point of multiplicityp≥1 of the nonlinear systemF(x, λ)=0,F:ℝⁿ×ℝ¹→ℝⁿ, if rank∂ _xF(x*, λ*)=n−1, and if the Ljapunov-Schmidt reduced equation has the normal formg(ξ, μ)=±ξ ²⁺ p±μξ=0. It is shown that such points satisfy a minimally extended systemG(y)=0,G:ℝ ⁿ⁺²→ℝⁿ⁺² the dimensionn+2 of which is independent ofp. For solving this system, a two-stage Newton-type method is proposed. Some numerical tests show the influence of the starting point and of the bordering vectors used in the definition of the extended system on the behavior of the iteration.     

Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u'))' + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)(p(t)u₁), …, q(t)(p(t)u_n)), where u = (u₁, …, u_n), andcovers the two important cases (u) = u and (u) = up > 1, h(t) = diag[h₁(t), …, h_n(t)] and f(u) = (f¹(u), …, fⁿ (u)). Assume that fⁱ and h_i are nonnegative continuous. For u = (u₁, …, u_n), let

, f₀ = maxf¹₀, …, fⁿ₀ and f_∞ = maxf¹_∞, …, fⁿ_∞. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.     

Verification of Reduced Convergence Rates

In this short article, we recalculate the numerical example in Kíek and Neittaanmäki (1987) for the Poisson solution u=x(1–x)siny in the unit square S as . By the finite difference method, an error analysis for such a problem is given from our previous study by where h is the meshspacing of the uniform square grids used, and C₁ and C₂ are two positive constants. Let =u–u_h, where u_h is the finite difference solution, and is the discrete H¹ norm. Several techniques are employed to confirm the reduced rate of convergence, and to give the constants, C₁=0.09034 and C₂=0.002275 for a stripe domain. The better performance for arises from the fact that the constant C₁ is much large than C₂, and the h in computation is not small enough.     

Performance Rating via the Feast Indices

Based on algorithmic and computational studies, we present the Feast Indices which are new indicators for the realistic performance of many recent processors, as typically used in ‘low cost’ PC's/workstations up to (parallel) supercomputers. Since these tests are very specifically designed for the evaluation of modern numerical simulation techniques, with a special emphasis on `large scale' FEM computations and iterative solvers of Krylov-space/multigrid type, they examine new aspects in comparison to the standard benchmark tests (LINPACK, SPEC FP95, NAS, STREAM) and allow new qualitative and particularly quantitative ratings of the various hardware platforms. We explore the computational efficiency of certain Linear Algebra components, hereby applying the typical sparse approaches and additionally the sparse banded style from Feast which enables us to exploit a significant percentage of the available Peak performance. The tests include ‘simple’ matrix-vector applications as well as complete multigrid solvers with very robust smoothers which are necessary for the efficient treatment of highly adapted meshes. Received: February 10, 1999; revised: June 14, 1999