Totally Discrete Explicit and Semi-implicit Euler Methods for a Blow-up Problem in Several Space Dimensions

The equation u _t =Δu+u ^p with homogeneous Dirichlet boundary conditions has solutions with blow-up if p>1. An adaptive time-step procedure is given to reproduce the asymptotic behavior of the solutions in the numerical approximations. We prove that the numerical methods reproduce the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.     

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.     

Superconvergence of Discontinuous Galerkin Methods for Convection-Diffusion Problems

Some discontinuous Galerkin methods for the linear convection-diffusion equation −ε u″+bu′=f are studied. Based on superconvergence properties of numerical fluxes at element nodes established in some earlier works, e.g., Celiker and Cockburn in Math. Comput. 76(257), 67–96, 2007, we identify superconvergence points for the approximations of u or q=u′. Our results are twofold: 1) For the minimal dissipation LDG method (we call it md-LDG in this paper) using polynomials of degree p, we prove that the leading terms of the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p+1, respectively. Consequently, the zeros of the right-Radau and left-Radau polynomials of degree p+1 are the superconvergence points of order p+2 for the discretization errors of the potential and of the gradient, respectively.     

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.     

On the rate of convergence of 2-term recursions in ℝ d

LetG be a compact set in ℝ ^d d≥1,M=G×G andϕ:M →G a map inC ₃(M). Suppose thatϕ has a fixed pointξ, i.e.ϕ(ξ, ξ)=ξ. We investigate the rate of convergence of the iterationx ⁿ⁺²=φ(x ⁿ⁺¹,x ⁿ) withx ⁰,x ¹∈G andx ⁿ→ξ. Iff _n=Q‖x ⁿ−ξ‖ with a suitable norm and a constantQ depending onξ, an exact representation forf _n is derived. The error terms satisfyf _2m+1≍(ƒ_2m)^γ,f _2m+2≍(ƒ_2m+1)^2γ,m≥0, with 1<gg<2, andγ=γ(x ¹,x ⁰). According to our main result we have lim_n→∞{‖x ⁿ⁺²‖/(‖x ⁿ‖)²}=Q, 0<Q<∞. This paper constitutes an extension of a part of the author’s doctoral thesis realized under the direction of Prof. E. Wirsing and Prof. A. Peyerimhoff, University of Ulm (Germany).     

Approximating Local Averages of Fluid Velocities: The Stokes Problem

As a first step to developing mathematical support for finite element approximation to the large eddies in fluid motion we consider herein the Stokes problem. We show that the local average of the usual approximate flow field u ^h over radius δ provides a very accurate approximation to the flow structures of O(δ) or greater. The extra accuracy appears for quadratic or higher velocity elements and degrades to the usual finite element accuracy as the averaging radius δ→h (the local meshwidth). We give both a priori and a posteriori error estimates incorporating this effect. Received December 3, 1999; revised October 16, 2000     

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.     

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.     

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.     

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.     

Two-Level Additive Schwarz Preconditioners for the h-p Version of the Galerkin Boundary Element Method for 2-d Problems

We study two-level additive Schwarz preconditioners for the h-p version of the Galerkin boundary element method when used to solve hypersingular integral equations of the first kind, which arise from the Neumann problems for the Laplacian in two dimensions. Overlapping and non-overlapping methods are considered. We prove that the non-overlapping preconditioner yields a system of equations having a condition number bounded by   where H _i is the length of the i-th subdomain, h _i is the maximum length of the elements in this subdomain, and p is the maximum polynomial degree used. For the overlapping method, we prove that the condition number is bounded by   where δ is the size of the overlap and H=max _i H _i . We also discuss the use of the non-overlapping method when the mesh is geometrically graded. The condition number in that case is bounded by clog² M, where M is the degrees of freedom. Received October 27, 2000, revised March 26, 2001     

Time-local formulation and identification of implicit Volterra models by use of diffusive representation

We present a time-continuous identification method for nonlinear dynamic Volterra models of the form HX=f(u,X)+v with H, a causal convolution operator. It is mainly based on a suitable parameterization of H deduced from the so-called diffusive representation, which is devoted to state representations of integral operators. Following this approach, the complex dynamic nature of H can be summarized by a few numerical parameters on which the identification of the dynamic part of the model will focus. The method is validated on a physical numerical example.     

Output-Sensitive Algorithm for Computing β-Skeletons

The β-skeleton is a measure of the internal shape of a planar set of points. We get an entire spectrum of shapes by varying the parameter β. For a fixed value of β, a β-skeleton is a geometric graph obtained by joining each pair of points whose β-neighborhood is empty. For β≥1, this neighborhood of a pair of points p _i ,p _j is the interior of the intersection of two circles of radius , centered at the points (1−β/2)p _i +(β/2)p _j and (β/2)p _i +(1−β/2)p _j , respectively. For β∈(0,1], it is the interior of the intersection of two circles of radius , passing through p _i and p _j . In this paper we present an output-sensitive algorithm for computing a β-skeleton in the metrics l ₁ and l _∞ for any β≥2. This algorithm is in O(nlogn+k), where k is size of the output graph. The complexity of the previous best known algorithm is in O(n ^5/2logn) [7]. Received April 26, 2000     

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.     

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)     

Constructing Labeling Schemes through Universal Matrices

Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs ℱ labels the vertices of all graphs in ℱ such that for every graph G∈ℱ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ℱ. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes.     

Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det (D ² u ⁰)=f (>0) based on the vanishing moment method which was developed by the authors in Feng and Neilan (J. Sci. Comput. 38:74–98, 2009) and Feng (Convergence of the vanishing moment method for the Monge-Ampère equation, submitted). In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation −εΔ² u ^ε +det D ² u ^ε =f accompanied by appropriate boundary conditions. This new approach enables us to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation (and other fully nonlinear second order partial differential equations), a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution u ^ε of the regularized problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter ε, for the error u^e-u^e_hu^{\varepsilon}-u^{\varepsilon}_{h}. Due to the strong nonlinearity of the underlying equation, the standard error estimate technique, which has been widely used for error analysis of finite element approximations of nonlinear problems, does not work here. To overcome the difficulty, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its finite element approximations. Finally, using the Argyris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of ε for the error u⁰-u_h^eu^{0}-u_{h}^{\varepsilon}, and numerically examine what is the “best” mesh size h in relation to ε in order to achieve these rates.     

Multiresolution Visualization of Higher Order Adaptive Finite Element Simulations

h –p–adaptive projection with respect to any prescribed threshold value for the visual error. This projection can then be processed by various local rendering methods, e.g. color coding of data or isosurface extraction. Especially for color coding purposes modern texture capabilities are used to directly render higher polynomial data by superposition of polynomial basis function textures and final color look-up tables. Numerical experiments from CFD clearly demonstrate the applicability and efficiency of our approach. Received September 25, 2001; revised March 31, 2003 Published online: May 26, 2003 The authors acknowledge the valuable hints of the anonymous referees.     

(Φ,Φ<Superscript>*</Superscript>) Image Decomposition Models and Minimization Algorithms

We propose in this paper minimization algorithms for image restoration using dual functionals and dual norms. In order to extract a clean image u from a degraded version f=Ku+n (where f is the observation, K is a blurring operator and n represents additive noise), we impose a standard regularization penalty Φ(u)=∫ φ(|Du|)dx<∞ on u, where φ is positive, increasing and has at most linear growth at infinity. However, on the residual f−Ku we impose a dual penalty Φ^*(f−Ku)<∞, instead of the more standard fidelity term. In particular, when φ is convex, homogeneous of degree one, and with linear growth (for instance the total variation of u), we recover the (BV,BV ^*) decomposition of the data f, as suggested by Y. Meyer (Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, Am. Math. Soc., Providence, 2001). Practical minimization methods are presented, together with theoretical, experimental results and comparisons to illustrate the validity of the proposed models. Moreover, we also show that by a slight modification of the associated Euler-Lagrange equations, we obtain well-behaved approximations and improved results.