A higher order richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation

The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform grids condensing in the vicinity of boundary layers converges ɛ-uniformly with an order at most almost one. The Richardson technique is used to construct a nonlinear scheme that converges ɛ-uniformly with an improved order, namely, at the rate O(N ₁⁻²ln² N ₁ + N ₂⁻²), where N ₁ + 1 and N ₂ + 1 are the number of grid nodes along the x ₁-axis and per unit interval of the x ₂-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct a linearized iterative Richardson scheme converging ɛ-uniformly with an improved order. Both the basic and improved iterative schemes converge ɛ-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration step at which the same ɛ-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown that no Richardson schemes exist for the convection-diffusion boundary value problem converging ɛ-uniformly with an order greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based.     

Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method

In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction-diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order derivatives converge in the maximum norm uniformly with respect to a perturbation parameter ɛ ∈(0, 1]; the normalized derivatives are ɛ-uniformly bounded. The key idea of this approach to the construction of ɛ-uniformly convergent finite difference schemes is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first- and second-order derivatives converge ɛ-uniformly at the rate of O(N ⁻²ln² N), where N + 1 is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed such that its solution and its normalized first and second derivatives converge ɛ-uniformly in the maximum norm at the same rate of O(N ⁻⁴ln⁴ N).     

Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity

The initial-boundary value problem in a domain on a straight line that is unbounded in x is considered for a singularly perturbed reaction-diffusion parabolic equation. The higher order derivative in the equation is multiplied by a parameter ɛ², where ɛ ∈ (0, 1]. The right-hand side of the equation and the initial function grow unboundedly as x → ∞ at a rate of O(x ²). This causes the unbounded growth of the solution at infinity at a rate of O(Ψ(x)), where Ψ(x) = x ² + 1. The initialboundary function is piecewise smooth. When ɛ is small, a boundary and interior layers appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristics of the reduced equation passing through the discontinuity points of the initial function. In the problem under examination, the error of the grid solution grows unboundedly in the maximum norm as x → ∞ even for smooth solutions when ɛ is fixed. In this paper, the proximity of solutions of the initial-boundary value problem and its grid approximations is considered in the weighted maximum norm ∥·∥ ^w with the weighting function Ψ⁻¹(x); in this norm, the solution of the initial-boundary value problem is ɛ-uniformly bounded. Using the method of special grids that condense in a neighborhood of the boundary layer or in neighborhoods of the boundary and interior layers, special finite difference schemes are constructed and studied that converge ɛ-uniformly in the weighted norm. It is shown that the convergence rate considerably depends on the type of nonsmoothness in the initial-boundary conditions. Grid approximations of the Cauchy problem with the right-hand side and the initial function growing as O(Ψ(x)) that converge ɛ-uniformly in the weighted norm are also considered.     

Diffusion approximation for the convection-diffusion equation with random drift

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂ _t u _ɛ (t, x) = κΔ _x (t, x) + 1/ɛV(t/ɛ²,xɛ) ·∇ _x u _ɛ (t, x) with the initial condition u _ɛ(0,x) = u ₀(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R ^d is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u _ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain constant coefficient heat equation. Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001     

A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces

Given a domain Ω of class C ^k,1, k ∈ ℕ, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of in the sense that (∂/∂x _n )α(x′, 0) = − N(x′) and that still is of class C ^k,1. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to k on domains of class C ^k,1. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.     

Multiscale stochastic homogenization of convection-diffusion equations

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ∂u _ɛ ^ω / ∂t+1 / ɛ ₃ C(T ₃(x/ɛ ₃)ω ₃) · ∇u _ɛ ^ω − div(α(T ₂(x/ɛ ₂)ω ₂, t) ∇u _ɛ ^ω ) = f. It is shown, under certain structure assumptions on the random vector field C(ω ₃) and the random map α(ω ₁, ω ₂, t), that the sequence {u _ɛ ^ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f).     

Fourier Analysis of Semistable Distributions

We consider the generalized convolution powers G _α ^*u (x) of an arbitrary semistable distribution function G _α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G _α ^(k,j)(x;u)=∂ ^k+j G _α ^*u (x)/∂ x ^k ∂ u ^j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at     

Singularly perturbed elliptic boundary value problems

Summary The paper treats elliptic operators of the form L(ɛ∂₁, ..., ɛ∂_n), where L is a polynomial in a variables of order 2m₁, and ɛ is a small parameter. Solutionsu _ɛ of Lu=0 in a half space satisfyng conditions B_j(ɛ∂₁, ɛ∂₂, ..., ɛ∂_n)u=ɛγ_jϕ_j(x)(j=1, ..., m₁) on the boundary are constructed and estimated using H?lder norms, Poisson kernels, and an elaborate potential theory. Properties of the interior limit u₀=u _ɛ(κ) are studied. The paper is preparatory to a detailed investigation of Schauder estimates for such problems with variables coefficients. Supported in part by N. S. F. Grant GP-11660. Entrata in Redazione il 9 gennaio 1971.     

A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation

For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ɛ, where ɛ ∈ (0, 1], the grid approximation of the Dirichlet problem on a rectangular domain in the (x, t)-plane is examined. For small ɛ, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this domain. A new approach to the construction of ɛ-uniformly converging difference schemes of higher accuracy is developed for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform grids. The base scheme converges ɛ-uniformly in the maximum norm at the rate of O(N ⁻²ln² N + N ₀⁻¹), where N + 1 and N ₀ + 1 are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme convergesɛ-uniformly at the rate of O(N ⁻⁴ln⁴ N + N ₀⁻²). For fixed values of the parameter, the convergence rate is O(N ⁻⁴ + N ₀⁻²).     

Solution to the boundary blowup problem for k-curvature equation

We consider the boundary blowup problem for k-curvature equation, i.e., H _k [u] = f(u) g(|Du|) in an n-dimensional domain Ω, with the boundary condition u(x) → ∞ as dist (x,∂Ω) → 0. We prove the existence result under some hypotheses. We also establish the asymptotic behavior of a solution near the boundary ∂Ω. Mathematics Subject Classification (2000) 35J65, 35B40, 53C21     

A finite difference scheme of improved accuracy on a priori adapted grids for a singularly perturbed parabolic convection-diffusion equation

In the case of the Dirichlet problem for a singularly perturbed parabolic convection-diffusion equation with a small parameter ɛ multiplying the higher order derivative, a finite difference scheme of improved order of accuracy that converges almost ɛ-uniformly (that is, the convergence rate of this scheme weakly depends on ɛ) is constructed. When ɛ is not very small, this scheme converges with an order of accuracy close to two. For the construction of the scheme, we use the classical monotone (of the first order of accuracy) approximations of the differential equation on a priori adapted locally uniform grids that are uniform in the domains where the solution is improved. The boundaries of such domains are determined using a majorant of the singular component of the grid solution. The accuracy of the scheme is improved using the Richardson technique based on two embedded grids. The resulting scheme converges at the rate of O((ɛ⁻¹ N ^−K ln² N)² + N ⁻²ln⁴ N + N ₀⁻²) as N, N ₀ → ∞, where N and N ₀ determine the number of points in the meshes in x and in t, respectively, and K is a prescribed number of iteration steps used to improve the grid solution. Outside the σ-neighborhood of the lateral boundary near which the boundary layer arises, the scheme converges with the second order in t and with the second order up to a logarithmic factor in x; here, σ = O(N ^{−(K − 1)}ln² N). The almost ɛ-uniformly convergent finite difference scheme converges with the defect of ɛ-uniform convergence ν, namely, under the condition N ⁻¹ ≪ ɛ^ν, where ν determining the required number of iteration steps K (K = K(ν)) can be chosen sufficiently small in the interval (0, 1]. When ɛ⁻¹ = O(N ^{K − 1}), the scheme converges at the rate of O(N ⁻²ln⁴ N + N ₀⁻²).     

A Fully Nonlinear Nonhomogeneous Neumann Problem

Let Ω be an open bounded set in ℝ^N, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|^p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative ∂/∂ν_a related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝ^N, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere defined in ℝ, with β(0)=γ(0)=0, f∈L¹(ℝ^N), g∈L¹(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,

A generalization of the Gagliardo inequality

Consider functions u₁, u₂,..., u_n ∈ D(ℝ^k) and assume that we are given a certain set of linear combinations of the form ∑_{i, j} a _ij ^(l) ∂_ju_i. Sufficient conditions in terms of coefficients a _ij ^(l) are indicated under which the norms are controlled in terms of the L¹-norms of these linear combinations. These conditions are mostly transparent if k = 2. The classical Gagliardo inequality corresponds to a single function u₁ = u and the collection of its partial derivatives ∂₁u,..., ∂_ku. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 120–139.     

Large solutions for equations involving the p-Laplacian and singular weights

In this paper we consider the boundary blow-up problem Δ_pu = a(x)u^q in a smooth bounded domain Ω of \mathbbR^N{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here D_pu = div(|?u|^p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.     

Asymptotic behaviour of a Brownian motion on exterior domains

We study the asymptotic behaviour of the transition density of a Brownian motion in ?, killed at ∂?, where ? ^c is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density p ^? _t (x, y) behaves, for large t, as u(x)u(y)(t(log t)²)⁻¹ for x, y∈?, where u is the unique positive harmonic function vanishing on (∂?) ^r , such that u(x) ∼ log ∣x∣. Received: 29 January 1999 / Revised version: 11 May 1999     

Scale limit of a variational inequality modeling diffusive flux in a domain with small holes and strong adsorption in case of a critical scaling

In this paper we study the asymptotic behavior of solutions u _ɛ of the elliptic variational inequality for the Laplace operator in domains periodically perforated by balls with radius of size C ₀ɛ_α, C ₀ > 0, α = n/n−2, and distributed with period ɛ. On the boundary of balls, we have the following nonlinear restrictions u _ɛ ≥ 0, ∂_ν u _ɛ ≥ −ɛ^−ασ(x, u _ɛ), u _ɛ(∂_ν u _ɛ + ɛ^−ασ(x, u _ɛ)) = 0. The weak convergence of the solutions u _ɛ to the solution of an effective variational equality is proved. In this case, the effective equation contains a nonlinear term which has to be determined as solution of a functional equation. Furthermore, a corrector result with respect to the energy norm is given.     

Homogenization of some parabolic operators with several time scales

The main focus in this paper is on homogenization of the parabolic problem ∂ _t uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ ^r )∇u ^ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.     

Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires

Résumé Soit X un processus gaussien stationnaire non dérivable. Nous étudions le nombre de passages en zéro du processus régularisé par convolution. Sous des hypothèses peu restrictives sur X, cette variable convenablement normalisée, converge au sens de L ² quand la taille du filtre tend vers zéro. Lorsque X admet un temps local continu, la limite obtenue est le temps local.

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Summary Let {X(t)} be a stationary non differentiable Gaussian process and let ϕ_ɛ(u=ɛ⁻¹ ϕ(u/ɛ) be an approximate identity. Setting X _ɛ(t)=X*ϕ_ɛ(t) and letting N _ɛ(T) be the number of zeros of X _ɛ in the interval [0, T] it is shown that under weak technical conditions there are constants C(ɛ) so that C(ɛ) N _ɛ(T) converges in L ² as ɛ→0. When X admits a continuous local time, the limit is the local time L(0, T) at zero of X(t).

     

Large solutions of elliptic equations with a weakly superlinear nonlinearity

This paper studies the asymptotic behavior near the boundary for large solutions of the semilinear equation Δu + au = b(x)f(u) in a smooth bounded domain Ω of ℝ^N with N ≥ 2, where a is a real parameter and b is a nonnegative smooth function on . We assume that f(u) behaves like u(ln u)^α as u → ∞, for some α > 2. It turns out that this case is more difficult to handle than those where f(u) grows like u ^p (p > 1) or faster at infinity. Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. We also obtain some uniqueness results. Research of both authors supported by the Australian Research Council.     

Quasilinear elliptic equations with boundary blow-up

Assume that Ω is a bounded domain in ℝ _N withN ≥2, which has aC ²-boundary. We show that forp ∃ (1, ∞) there exists a weak solutionu of the problem δ_p u(x) = f(u(x)), x ∃ Ω with boundary blow-up, wheref is a positive, increasing function which meets some natural conditions. The boundary blow-up ofu(x) is characterized in terms of the distance ofx from ∂Ω. For the Laplace operator, our results coincide with those of Bandle and Essén [1]. Finally, for a rather wide subclass of the class of the admissible functionsf, the solution is unique whenp ∃ (1, 2].