A note on initial higher‐order convergence results for boundary element methods with approximated boundary conditions

We consider a variational problem associated with a pseudo‐differential operator of negative order 2s < 0 with an additional approximation of the given linear form. Such an approximation may correspond to an interpolation of given boundary conditions for a partial differential equation. The asymptotic order of convergence of the related Galerkin solution can be reached for ν = μ +2s, where ν and μ are the polynomial degrees of the trial functions used to approximate the solution and boundary conditions, respectively. The main result of this article is to prove that one can expect higher initial rates in the convergence behavior, even in the worst case of isoparametric approximations (ν = μ) when the error is measured in the Sobolev norm H^τ(Γ) with τ ∈ [s, 0]; i.e., this initial estimate is also valid in the energy norm ‖ · ‖. This result is based on the relation between the approximation error of the Galerkin solution without this additional approximation and the additional approximation error itself. As an illustration of the technique, an application of a boundary element method for the Dirichlet problem of a second‐order elliptic partial differential operator is given. Numerical examples confirm the theoretical results for this case. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 581–588, 2000.     

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

In this paper, we establish some sharp Sobolev trace inequalities on n-dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2(n - 1)/(n - 2), 1/S = inf {∫ |∇u|² : ∇u ∈ L²(R₊ⁿ), ∫ |u|^q = 1}. We establish for any Riemannian manifold with a smooth boundary, denoted as (M, g), that there exists some constant A = A(M, g) > 0, (∫_dM|u|^q ds_g)^2/q < or = to S ∫_M |∇_gu|² dv_g + A ∫_dMu² ds_g, for all u ∈ H¹ (M). The inequality is sharp in the sense that the inequality is false when S is replaced by any smaller number. © 1997 John Wiley & Sons, Inc.     

Uniform stability of spectral nonlinear Galerkin methods

This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier‐Stokes equations with some appropriate assumption of the data (λ, u₀, f). If the backward Euler scheme with the semi‐implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δt ≤ (2/λλ₁). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δt = O(λ) and Δt = O(λ), respectively, where λ ≤ λ, which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Finding blocks and other patterns in a random coloring of ℤ

Let ξ = (ξ_k)_k∈? be i.i.d. with P(ξ_k = 0) = P(ξ_k = 1) = 1/2, and let S: = (S_k) be a symmetric random walk with holding on ?, independent of ξ. We consider the scenery ξ observed along the random walk path S, namely, the process (χ_k := ξ). With high probability, we reconstruct the color and the length of blockⁿ, a block in ξ of length ≥ n close to the origin, given only the observations (χ_k). We find stopping times that stop the random walker with high probability at particular places of the scenery, namely on blockⁿ and in the interval [?3ⁿ,3ⁿ]. Moreover, we reconstruct with high probability a piece of ξ of length of the order 3 around blockⁿ, given only 3 observations collected by the random walker starting on the boundary of blockⁿ. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Superconvergence of a 3D finite element method for stationary Stokes and Navier‐Stokes problems

For the Poisson equation on rectangular and brick meshes it is well known that the piecewise linear conforming finite element solution approximates the interpolant to a higher order than the solution itself. In this article, this type of supercloseness property is established for a special interpolant of the Q₂ ? P element applied to the 3D stationary Stokes and Navier‐Stokes problem, respectively. Moreover, applying a Q₃ ? P postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q₂‐interpolation without influencing the superconvergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost‐benefit analysis between the two third‐order methods, the post‐processed Q₂ ? P discretization, and the Q₃ ? P discretization is carried out. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Robust characterizations of k‐wise independence over product spaces and related testing results

A discrete distribution D over Σ₁ ×··· ×Σ_n is called (non‐uniform) k ‐wise independent if for any subset of k indices {i₁,…,i_k} and for any z₁∈Σ,…,z_k∈Σ, Pr_X～D[X···X = z₁···z_k] = Pr_X～D[X = z₁]···Pr_X～D[X = z_k]. We study the problem of testing (non‐uniform) k ‐wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k ‐wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}ⁿ. For the non‐uniform case, we give a new characterization of distributions being k ‐wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496–505) on uniform k ‐wise independence over the Boolean cubes to non‐uniform k ‐wise independence over product spaces. Our results yield natural testing algorithms for k ‐wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Law of large numbers for increasing subsequences of random permutations

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1,…,n}. We show that Var(Z) = o((EZ)²) as n → ∞ if and only if . In particular then, the weak law of large numbers holds for Z if ; that is, We also show the following approximation result for the uniform measure U_n on S_n. Define the probability measure μ on S_n by where U denotes the uniform measure on the subset of permutations that contain the increasing subsequence {x₁,x₂,…,x}. Then the weak law of large numbers holds for Z if and only if where ∣∣˙∣∣ denotes the total variation norm. In particular then, (*) holds if . In order to evaluate the asymptotic behavior of the second moment, we need to analyze occupation times of certain conditioned two‐dimensional random walks. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=D_νu|=θ|=0 and initial condition: (u, u_t, θ)|_t=0=(u₀, v₀, θ₀). Here, Ω is a bounded domain in ?ⁿ(n≧2). We assume that the boundary ?Ω of Ω is a C⁴ hypersurface. We obtain an L_p–L_q maximal regularity theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

About smoothness of solutions of the heat equations in closed,smooth space‐time domains

We consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ?² = {(x¹, x²)} of class C^2k. We give sufficient conditions for u to have k^th‐order continuous derivatives with respect to (x¹, x²) in D? for integers k ≥ 2. The equation is supplemented with C^2k boundary data, and we assume that f ? C^2(k?1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc.     

A two‐level method in space and time for the Navier‐Stokes equations

A two‐level method in space and time for the time‐dependent Navier‐Stokes equations is considered in this article. The approximate solution u_M∈H_M is decomposed into the large eddy component v∈H_m(m < M) and the small eddy component w∈H. We obtain the large eddy component v by solving a standard Galerkin equation in a coarse‐level subspace H_m with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H with a time step length pk, where p is a positive integer. The analysis shows that our two‐level scheme has long‐time stability and can reach the same accuracy as the standard Galerkin method in fine‐level subspace H_M for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Star factorizations of graph products

A k‐star is the graph K_1,k. We prove a general theorem about k‐star factorizations of Cayley graphs. This is used to give necessary and sufficient conditions for the existence of k‐star factorizations of any power (K_q)^s of a complete graph with prime power order q, products C × C ×··· × C of k cycles of arbitrary lengths, and any power (C_r)^s of a cycle of arbitrary length. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 59–66, 2001     

Strong solutions to the Keller‐Segel system with the weak Ln /2 initial data and its application to the blow‐up rate

We shall show an exact time interval for the existence of local strong solutions to the Keller‐Segel system with the initial data u₀ in L^{n /2}_w (?ⁿ), the weak L^{n /2}‐space on ?ⁿ. If ‖u₀‖ is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in L^{n /2}_w (?ⁿ) stems from obtaining a self‐similar solution which does not belong to any usual L^p(?ⁿ). Furthermore, the characterization of local existence of solutions gives us an explicit blow‐up rate of ‖u (t)‖ for n /2 < p < ∞ as t → T_max, where T_max denotes the maximal existence time (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space‐time for the two‐dimensional nonstationary Navier‐Stokes problem is considered. The method is a multi‐scale method in which the fully nonlinear Navier‐Stokes problem is only solved on a single coarsest space‐time mesh; subsequent approximations are generated on a succession of refined space‐time meshes by solving a linearized Navier‐Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J‐level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: h_j ～ h, k_j ～ k, j = 2, …, J, the J‐level finite element method in space‐time provides the same accuracy as the one‐level method in space‐time in which the fully nonlinear Navier‐Stokes problem is solved on a final finest space‐time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Superconvergence of interpolated collocation solutions for Hammerstein equations

In this article, we discuss the superconvergence of the interpolated collocation solutions for Hammerstein equations. Applying this new interpolation postprocessing to the collocation approximation x_h, we get a higher accuracy approximation I x_h, whose convergence order is the same as that of the iterated collocation method. Such an interpolation postprocessing method is much simpler. Also, numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On the convergence rate of vanishing viscosity approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) ? u^?(t, ·)‖ = O(1)(1 + t) · |ln ?| on the distance between an exact BV solution u and a viscous approximation u^?, letting the viscosity coefficient ? → 0. In the proof, starting from u we construct an approximation of the viscous solution u^? by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ?. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.     

Finite element method by using quartic B‐splines

In this article, we discuss a kind of finite element method by using quartic B‐splines to solve Dirichlet problem for elliptic equations. Bivariate spline proper subspace of S(Δ) satisfying homogeneous boundary conditions on Type‐2 triangulations and quadratic B‐spline interpolating boundary functions are primarily constructed. Linear and nonlinear elliptic equations are solved by Galerkin quartic B‐spline finite element method. Numerical examples are provided to illustrate the proposed method is flexible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 818–828, 2011     

Convergence of boundary optimal controls in mixed elliptic problems

We consider a steady-state heat conduction problem P_α withmixed boundary conditions for the Poisson equation in a bounded multidimensional domain Ω depending of a positive parameter α which represents the heat transfer coefficient on a portion Γ₁ of the boundary of Ω. We consider, for each α > 0, a cost function J_α and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion Γ₂ of the boundary of Ω. We obtain that the optimality conditions are given by a complementary free boundary problem in Γ₂ in terms of the adjoint state. We prove that the optimal control q and its corresponding system state u and adjoint state p for each α are strongly convergent to q_op, u and p in L²(Γ₂), H¹(Ω), and H¹(Ω) respectively when α → ∞. We also prove that these limit functions are respectively the optimal control, the system state and the adjoint state corresponding to another boundary optimal control problem with restrictions for the same Poisson equation with a different boundary condition on the portion Γ₁. We use the elliptic variational inequality theory in order to prove all the strong convergences. In this paper, we generalize the convergence result obtained in Ben Belgacem-El Fekih-Metoui, ESAIM:M2AN, 37 (2003), 833-850 by considering boundary optimal control problems with restrictions on the heat flux q defined on Γ₂ and the parameter α (which goes to infinity) is defined on Γ₁. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Obstacle Problems with Linear Growth: Hölder Regularity for the Dual Solution

For a strictly convex integrand f : ℝⁿ → ℝ with linear growth we discuss the variational problem among mappings u : ℝⁿ ⊃ Ω → ℝ of Sobolev class W¹₁ with zero trace satisfying in addition u ≥ ψ for a given function ψ such that ψ|_∂Ω < 0. We introduce a natural dual problem which admits a unique maximizer σ. In further sections the smoothness of σ is investigated using a special J-minimizing sequence with limit u* ∈ C^1,α (Ω) for which the duality relation holds.     

The existence of positive solutions to a non‐local singular boundary value problem

We consider the non‐local singular boundary value problem (1) where q ∈ C⁰([0,1]) and f, h ∈ C⁰((0,∞)), limf(x)=?∞, limh(x)=∞. We present conditions guaranteeing the existence of a solution x ∈ C¹([0,1]) ∩ C²((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.