### A C...-Collocation-like Method for Two-Point Boundary Value Problems.

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A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation $-\u03f5{u}^{n}+p{u}^{\text{'}}+qu=f$ are presented and analyzed theoretically. The positive number $\u03f5$ is supposed to be much less than the discretization step and the values of $\left|p\right|,q$. An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.

FEM discretizations of arbitrary order $r$ are considered for a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers. A posteriori error bounds of interpolation type are derived in the maximum norm. An adaptive algorithm is devised to resolve the boundary layers. Numerical experiments complement our theoretical results.

We study a Sturm-Liouville problem containing a spectral parameter in the boundary conditions. We associate to this problem a self-adjoint operator in a Pontryagin space Π₁. Using this operator-theoretic formulation and analytic methods, we study the asymptotic behavior of the eigenvalues under the variation of a large physical parameter in the boundary conditions. The spectral analysis is applied to investigate the well-posedness and stability of the wave equation of a string.