Stable cheapest nonconforming finite elements for the Stokes equations

We introduce two pairs of stable cheapest nonconforming finite element space pairs to approximate the Stokes equations. One pair has each component of its velocity field to be approximated by the $P_1$ nonconforming quadrilateral element while the pressure field is approximated by the piecewise constant function with globally two-dimensional subspaces removed: one removed space is due to the integral mean--zero property and the other space consists of global checker--board patterns. The other pair consists of the velocity space as the $P_1$ nonconforming quadrilateral element enriched by a globally one--dimensional macro bubble function space based on $DSSY$ (Douglas-Santos-Sheen-Ye) nonconforming finite element space; the pressure field is approximated by the piecewise constant function with mean--zero space eliminated. We show that two element pairs satisfy the discrete inf-sup condition uniformly. And we investigate the relationship between them. Several numerical examples are shown to confirm the efficiency and reliability of the proposed methods

Error Estimates for Approximations of Distributed Order Time Fractional Diffusion with Nonsmooth Data

In this work, we consider the numerical solution of an initial boundary value problem for the distributed order time fractional diffusion equation. The model arises in the mathematical modeling of ultra-slow diffusion processes observed in some physical problems, whose solution decays only logarithmically as the time $t$ tends to infinity. We develop a space semidiscrete scheme based on the standard Galerkin finite element method, and establish error estimates optimal with respect to data regularity in $L^2(D)$ and $H^1(D)$ norms for both smooth and nonsmooth initial data. Further, we propose two fully discrete schemes, based on the Laplace transform and convolution quadrature generated by the backward Euler method, respectively, and provide optimal convergence rates in the $L^2(D)$ norm, which exhibits exponential convergence and first-order convergence in time, respectively. Extensive numerical experiments are provided to verify the error estimates for both smooth and nonsmooth initial data, and to examine the asymptotic behavior of the solution.Comment: 25 pages, 2 figure

A class of nonparametric DSSY nonconforming quadrilateral elements

A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [J.Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming {G}alerkin methods based on quadrilateral elements for second order elliptic problems. ESAIM--Math. Model. Numer. Anal., 33(4):747--770, 1999]. The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence [Z. Cai, J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming quadrilateral finite elements: A correction. Calcolo, 37(4):253--254, 2000], while the new nonparametric DSSY elements require only four degrees of freedom. The design of new elements is based on the decomposition of a bilinear transform into a simple bilinear map followed by a suitable affine map. Numerical results are presented to compare the new elements with the parametric DSSY element.Comment: 20 page

Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems

Low-order nonconforming Galerkin methods will be analyzed for second-order elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P 1 , as for conforming elements; however, it is necessary to introduce new elements in the rectangular case. Optimal order error estimates are demonstrated in all cases with respect to a broken norm in H 1 (Ω) and in the Neumann and Robin cases in L 2 (Ω).Facultad de Ciencias Astronómicas y Geofísica

F. John's stability conditions vs. A. Carasso's SECB constraint for backward parabolic problems

In order to solve backward parabolic problems F. John [{\it Comm. Pure. Appl. Math.} (1960)] introduced the two constraints "$\|u(T)\|\le M$" and $\|u(0) - g \| \le \delta$ where $u(t)$ satisfies the backward heat equation for $t\in(0,T)$ with the initial data $u(0).$ The {\it slow-evolution-from-the-continuation-boundary} (SECB) constraint has been introduced by A. Carasso in [{\it SIAM J. Numer. Anal.} (1994)] to attain continuous dependence on data for backward parabolic problems even at the continuation boundary $t=T$. The additional "SECB constraint" guarantees a significant improvement in stability up to $t=T.$ In this paper we prove that the same type of stability can be obtained by using only two constraints among the three. More precisely, we show that the a priori boundedness condition $\|u(T)\|\le M$ is redundant. This implies that the Carasso's SECB condition can be used to replace the a priori boundedness condition of F. John with an improved stability estimate. Also a new class of regularized solutions is introduced for backward parabolic problems with an SECB constraint. The new regularized solutions are optimally stable and we also provide a constructive scheme to compute. Finally numerical examples are provided.Comment: 15 pages. To appear in Inverse Problem

Turing Instability for a Ratio-Dependent Predator-Prey Model with Diffusion

Ratio-dependent predator-prey models have been increasingly favored by field ecologists where predator-prey interactions have to be taken into account the process of predation search. In this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction-diffusion model in which predator mortality is neither a constant nor an unbounded function, but it is increasing with the predator abundance. We show that analytically at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion). We also show that the stationary solution becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises. A numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented. Numerical examples are also included