We review some numerical analysis of an adaptive finite element method (AFEM) for a class of elliptic partial differential equations based on a perturbation argument. This argument makes use of the relationship between the general problem and a model problem, whose adaptive finite element analysis is existing, from which we get the convergence and the complexity of adaptive finite element methods for a nonsymmetric boundary value problem, an eigenvalue problem, a nonlinear boundary value problem...

In this paper, a multi-parameter error resolution
technique is applied into a mixed finite element method for the
Stokes problem. By using this technique and establishing a multi-parameter
asymptotic error expansion for the mixed finite element method, an approximation of higher
accuracy is obtained by multi-processor computers in parallel.

Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing $N$ for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on $N\times \sqrt{N}$, $\sqrt{N}\times N$ and $\sqrt{N}\times \sqrt{N}$ meshes. It is shown that the combination FEM yields (up to a factor $lnN$) the same order of accuracy in the associated...

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