In the Ritz-Galerkin method the linear subspace of the trial solution is extended to a closed subset. Some results, such as orthogonalization and minimum property of the error function, are obtained. A second order scheme is developed for solving a linear singular perturbation elliptic problem and error estimates are given for a uniform mesh size. Numerical results for linear and semilinear singular perturbation problems are included.

Under an assumption of distribution on zeros of the polynomials, we have given the estimate of computational cost for the resultant method. The result in that, in probability $1-\mu$, the computational cost of the resultant method for finding $ε$-approximations of all zeros is at most $$cd^2(log d+log\frac{1}{\mu}+loglog\frac{1}ε)$$, where the cost is measured by the number of f-evaluations. The estimate of cost can be decreased to $c(d^2logd+d^2log\frac{1}{\mu}+dloglog\frac{1}ε)$ by combining resultant method with parallel quasi-Newton method.

We consider linear parabolic equations in a space-time domain with curved boundaries and nonhomogeneous Dirichlet boundary conditions and discuss their approximations with isoparametric space-time finite elements. A general error estimate is proved and applied to some elements of practical interest.

The exterior boundary value problems of Laplace equation and linear elastic equations are considered. A series of approximate infinite boundary conditions are given. Then the original problem is reduced to a boundary value problem on a bounded domain. The finite element approximation of this problem and its error estimate are obtained. Finally, a numerical example shows that this method is very effective.