In this paper, we consider the evaluation of flux integral outputs from reduced basis solutions to second-order PDE’s. In order to evaluate such outputs, a lifting function v must be chosen. In the standard finite element context, this choice is not relevant, whereas in the reduced basis context, as we show, it greatly affects the output error. We propose two “good” choices for v, and illustrat...

Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adju...

A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fin...

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion c...

A generalized tanh method has been proposed and discussed. This method has been employed for solving two well-known models of nonlinear equations, namely the Drinfeld-Sokolov (DS) system and PHI-four equation. The exact solutions of these models are obtained. Finally, some wave solutions are exhibited.

The homotopy analysis method (HAM) is used to find approximate analytical solutions of continuous population models for single and interacting species. The homotopy analysis method contains the auxiliary parameter ~, which provides us with a simple way to adjust and control the convergence region of series solution. the solutions are compared with the numerical results obtained using NDSolve, a...

We study the approximation properties of some general finiteelement spaces constructed using improved graded meshes. In our results, either the approximating function or the function to be approximated (or both) are in a weighted Sobolev space. The finite-element spaces that we define are obtained from conformally invariant families of finite elements (no affine invariance is used), stressing t...

In this paper, the optimal homotopy analysis method is applied to find the solitary wave solutions of the Kuramoto-Sivashinsky equation. With three auxiliary convergence-control parameters, whose possible optimal values can be obtained by minimizing the averaged residual error, the method used here provides us with a simple way to adjust and control the convergence region of the solution. Compa...

In this paper the general hybrid-mixed ®nite element methods are investigated systematically in a framework of multi-®eld variational equations. The commonly accepted concept ``saddle point problem'' is argued in this paper. The existence, uniqueness, convergence, and stability properties of the solutions are proved undertaking the assumptions of Ker*-ellipticity and nested BBconditions. The re...

In [7, 8], they proposed a new singular function(NSF) method to compute singular solutions of Poisson equations on a polygonal domain with re-entrant angles. Singularities are eliminated and only the regular part of the solution that is in H is computed. The stress intensity factor and the solution can be computed as a post processing step. This method was extended to the interface problem and ...