The maximum norm analysis of a nonmatching grids method for a class of parabolic $p(x)$-Laplacien equation

Motivated by the work of Boulaaras and Haiour in [7], we provide a maximum norm analysis of Schwarz alternating method for parabolic p(x)-Laplacien equation, where an optimal error analysis each subdomain between the discrete Schwarz sequence and the continuous solution of the presented problem is established

The Maximum Norm Analysis of Schwarz Method for Elliptic Quasi-Variational Inequalities

In this paper, we present a maximum norm analysis of an overlapping Schwartz method on non matching grids for a quasi-variational inequality, where the obstacle and the second member depend on the solution. Our result improves and generalizes some previous results.

Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique

Purpose A general strategy is developed for adaptive finite element (FE) analysis of free vibration of elastic membranes based on the element energy projection (EEP) technique. Design/methodology/approach By linearizing the free vibration problem of elastic membranes into a series of linear equivalent problems, reliable a posteriori point-wise error estimator is constructed via EEP super-convergent technique. Hierarchical local mesh refinement is incorporated to better deal with tough problems. Findings Several classical examples were analyzed, confirming the effectiveness of the EEP-based error estimation and overall adaptive procedure equipped with a local mesh refinement scheme. The computational results show that the adaptively-generated meshes reasonably catch the difficulties inherent in the problems and the procedure yields both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm. Originality/value By reasonable linearization, the linear-problem-based EEP technique is successfully transferred to two-dimensional eigenproblems with local mesh refinement incorporated to effectively and flexibly deal with singularity problems. The corresponding adaptive strategy can produce both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm and thus can be expected to apply to other types of eigenproblems.

Maximum Norm Estimates of the Solution of the Navier-Stokes Equations in the Halfspace with Bounded Initial Data

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in ℝ + n for n ≥ 3 with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in T n and ℝ n respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.

All finite sets are Ramsey in the maximum norm

For two metric spaces $\mathbb X$ and $\mathcal Y$ the chromatic number $\chi ({{\mathbb X}};{{\mathcal{Y}}})$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest k such that there is a colouring of the points of $\mathbb X$ with k colors that contains no monochromatic copy of $\mathcal Y$ . In this article, we show that for each finite metric space $\mathcal {M}$ that contains at least two points the value $\chi \left ({{\mathbb R}}^n_\infty; \mathcal M \right )$ grows exponentially with n. We also provide explicit lower and upper bounds for some special $\mathcal M$ .

Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Convection-Diffusion Equation

The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x-derivative in the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval {(0,1]}. For small ε, the problem involves a boundary layer of width {\mathcal{O}(\varepsilon)}, where the solution changes by a finite value, while its derivative grows unboundedly as ε tends to zero. We construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives). Using a priori estimates, we show that such a scheme converges as {\{\varepsilon N\},N_{0}\to\infty} in the maximum norm with first-order accuracy in {\{\varepsilon N\}} and {N_{0}}; as {N,N_{0}\to\infty}, the convergence is conditional with respect toN, where {N+1} and {N_{0}+1} are the numbers of mesh points in x and t, respectively. We develop an improved difference scheme on uniform meshes using the grid approximation of the first x-derivative in the convective term by the central difference operator under the condition{h\leq m\varepsilon}, which ensures the monotonicity of the scheme; here m is some rather small positive constant. It is proved that this scheme converges in the maximum norm at a rate of {\mathcal{O}(\varepsilon^{-2}N^{-2}+N^{-1}_{0})}. We compare the convergence rate of the developed scheme with the known Samarskii scheme for a regular problem. It is found that the improved scheme (for {\varepsilon=1}), as well as the Samarskii scheme, converges in the maximum norm with second-order accuracy inx and first-order accuracy int.