Mixed covolume method for parabolic problems on triangular grids

In this paper, we present a mixed covolume method for parabolic equations on triangular grids. This method use the lowest order Raviart–Thomas (R–T) mixed finite element space as the trial space. We prove the optimal order of convergence for the approximate pressure and velocity in L²-norm. Furthermore, we obtain the quasi-optimal error estimates for the approximate pressure in L^∞-norm.     

Remark on asymptotic behaviors of solutions to a quasi-linear elliptic Dirichlet problem with large diffusion

We study asymptotic behaviors of nontrivial solutions to the Dirichlet problem of a quasi-linear elliptic equation and obtain a lower bound for growth of L^∞-norm of the solutions, which implies the L^∞-norm of the solutions goes to infinity as the diffusion coefficient goes to infinity.     

The pointwise estimates of a conservative difference scheme for Burgers' equation

In this article, we are concerned with the numerical analysis of a nonlinear implicit difference scheme for Burgers' equation. A priori estimation of the analytical solution is provided in the sense of L^∞ -norm when the initial value is bounded in H¹-norm. Conservation, boundedness, and unique solvability are proved at length. Inspired by the method of the priori estimation for the analytical solution, we prove the convergence and stability of the difference scheme in L^∞ -norm. Finally, numerical examples are carried out to verify our theoretical results.     

A Maximum Principle for an Explicit Finite Difference Scheme Approximating the Hodgkin-Huxley Model

We analyze an explicit finite difference scheme for the general form of the Hodgkin-Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in excitable cells. We prove that the numerical solution is bounded in the L^∞-norm and L² converges to a unique solution. The L^∞-bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000) 65F20     

Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels

Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L²-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L²-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.     

Error estimate for finite volume scheme

We study the convergence of a finite volume scheme for the linear advection equation with a Lipschitz divergence-free speed in R ^d . We prove a h ^1/2-error estimate in the L ^∞(0,t;L ¹)-norm for BV data. This result was expected from numerical experiments and is optimal.     

Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

In this article, we propose and analyze several numerical methods for the nonlinear delay reaction–diffusion system with smooth and nonsmooth solutions, by using Quasi-Wilson nonconforming finite element methods in space and finite difference methods (including uniform and nonuniform L1 and L2-1_σ schemes) in time. The optimal convergence results in the senses of L²-norm and broken H¹-norm, and H¹-norm superclose results are derived by virtue of two types of fractional Grönwall inequalities. Then, the interpolation postprocessing technique is used to establish the superconvergence results. Moreover, to improve computational efficiency, fast algorithms by using sum-of-exponential technique are built for above proposed numerical schemes. Finally, we present some numerical experiments to confirm the theoretical correctness and show the effectiveness of the fast algorithms.     

Approximation of infinitely differentiable multivariate functions is intractable

We prove that L_∞-approximation of C^∞-functions defined on [0,1]^d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε(0,1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.     

On the compact difference scheme for the two-dimensional coupled nonlinear Schrödinger equations

A high-order finite difference method for the two-dimensional coupled nonlinear Schrödinger equations is considered. The proposed scheme is proved to preserve the total mass and energy in a discrete sense and the solvability of the scheme is shown by using a fixed point theorem. By converting the scheme in the point-wise form into a matrix–vector form, we use the standard energy method to establish the optimal error estimate of the proposed scheme in the discrete L²-norm. The convergence order is proved to be of a fourth-order in space and a second-order in time, respectively. Finally, some numerical examples are given in order to confirm our theoretical results for the numerical method. The numerical results are compared with exact solutions and other existing method. The comparison between our numerical results and those of Sun and Wangreveals that our method improves the accuracy of space and time directions.     

Sparsest solutions of underdetermined linear systems via -minimization for

We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal ℓ_q-quasinorm is also the sparsest one. This generalizes, and slightly improves, a similar result for the ℓ₁-norm. We then introduce a simple numerical scheme to compute solutions with minimal ℓ_q-quasinorm, and we study its convergence. Finally, we display the results of some experiments which indicate that the ℓ_q-method performs better than other available methods.     

Isotonic Approximation in L1

Let (Ω, ,P) be a measurable space and a sub-σ-lattice of the σ-algebra . For XL₁(Ω, ,P) we denote by P X the set of conditional 1-mean (or best approximants) of X given L₁( ) (the set of all -measurable and integrable functions). In this paper, we obtain characterizations of the elements in P X, similar to those obtained by Landers and Rogge for conditional s-means with 1<s<∞. Moreover, using these characterizations we can extend the operator P to a bigger space L₀(Ω, ,P). When, in certain sense, _n goes to _∞, we will be able to prove theorems about convergence and we will obtain bounds for the maximal function sup_nP _nX. A sharper characterization of conditional 1-means for certain particular σ-lattice was proved in previous papers. In the last section of this paper we generalize those results to all totally ordered σ-lattices.     

Convergence of greedy approximation for the trigonometric system

Summary We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form G_m(f ) := ∑_kЄΛ f^(k) e ^{(i k,x)}, where ΛˆZ^d is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients f^ (k) of function f . Note that G_m(f ) gives the best m-term approximant in the L₂-norm and, therefore, for each f ЄL₂, ║f-G_m(f )║₂→0 as m →∞. It is known from previous results that in the case of p ≠2 the condition f ЄL_p does not guarantee the convergence ║f-G_m(f )║_p→0 as m →∞.. We study the following question. What conditions (in addition to f ЄL_p) provide the convergence ║f-G_m(f )║_p→0 as m →∞? In our previous paper [10] in the case 2< p ≤∞ we have found necessary and sufficient conditions on a decreasing sequence {A_n}_n=1^∞ to guarantee the L_p-convergence of {G_m(f )} for all f ЄL_p , satisfying a_n (f ) ≤A_n , where {a_n (f )} is a decreasing rearrangement of absolute values of the Fourier coefficients of f. In this paper we are looking for necessary and sufficient conditions on a sequence {M (m)} such that the conditions f ЄL_p and ║G_M(m)(f ) - G_m(f )║_p →0 as m →∞ imply ║f - G_m(f )║_p →0 as m →∞. We have found these conditions in the case when p is an even number or p = ∞.     

On the accuracy of a difference scheme for solving the nonstationary coupled thermoelastic problem in stresses

For numerical solution of the coupled one-dimensional problem of dynamic thermoelasticity in stresses (strains) we construct a second-order approximating difference scheme. We study its stability and obtain an a priori estimate. We prove that the solution of the scheme converges to a generalized solution of the original problem in the Sobolev class W ₂ ² (Q_T).Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 95–99.     

关于求解非线性耦合Schrodinger 方程的 Sonnier - Christov 格式

该文对求解非线性耦合Schrodinger方程的Sonnier-Christov格式进行了数值分析, 证明了格式关于L₂范数的稳定性和二阶收敛性, 运用Brouwer不动点定理证明了差分解的存在唯一性, 给出一个求解非线性差分方程组的迭代算法并证明了算法的收敛性, 最后对双孤立波的碰撞进行了模拟.     

Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data

We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton–Jacobi–Bellman equation of the form v _t + max _α (f(x, α)v _x ) = 0, v(0, x) = v ₀(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1–33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L ¹-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.     

L 1-convergence of double cosine- and Walsh-Fourier series

We prove the convergence inL ¹([−gp, π)²)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa _jk satisfying certain conditions of Hardy-Karamata kind, and such thata _jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL ¹ (0, 1)²)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesàro summable. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # 234.     

On homogenization of periodic parabolic systems

We study homogenization in the small period limit for a periodic parabolic Cauchy problem in ^d and prove that the solutions converge in L ₂(^d) to the solution of the homogenized problem for each t > 0. For the L₂(^d)-norm of the difference, we obtain an order-sharp estimate uniform with respect to the L ₂(^d)-norm of the initial value.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 86–90, 2004Original Russian Text Copyright © by T. A. SuslinaSupported by RFBR grant No. 02-01-00798.     

On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations

We prove two sufficient conditions for local regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. One of these conditions implies the smoothness of L_3,∞-solutions as a particular case. Bibliography: 12 titles.Dedicated to Vsevolod Alekseevich Solonnikov__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 186–198.     

Littlewood–Paley and Multiplier Theorems on Weighted Spaces over Locally Compact Vilenkin Groups

We extend the Littlewood–Paley theorem toL^p_w(G), whereGis a locally compact Vilenkin group andware weights satisfying the MuckenhouptA_pcondition. As an application we obtain a mixed-norm type multiplier result onL^p_w(G) and prove the sharpness of our result. We also obtain a sufficient condition for φ L^∞(Γ) to be a multiplier on the power weightedL^p_α(G) in terms of its smoothness condition.     

An error estimate uniform in time for spectral semi-galerkin approximations of the nonhomogeneous navier-stokes equations

We consider the spectral semi-Galerkin method applied to the non-homogeneous Navier-Stokes equations, which describes the motion of miscibles fluids. Under certain conditions it is known that the aproximate solutions constructed by using this method converge to a global strong solution of these equations. In this paper we prove that these solutions satisfy an optimal uniform in time error estimate in the H ¹-norm for the velocity. We also derive an uniform error estimate in the L ^∞-norm for the density and an improved error estimate in the L ²-norm for the velocity.