Analytical and numerical methods for boundary value problems of mixed type

Abstract

Existence of a weak solution for some boundary value problems of mixed and hyperbolic types is proved by constructing multipliers which satisfy the multiplier inequalities. The nature of a singularity is then examined by utilizing norm estimates obtained by multipliers. A weak solution of a model hyperbolic problem is shown to have, in an integral sense, a logarithmic singularity at one parabolic point. A model mixed-type problem exhibits a fundamental-type singularity, when solved numerically. In general, it is very difficult to construct multipliers for an arbitrary mixed-type equation and an arbitrary domain. The multipliers constructed in this study satisfy the multiplier inequalities only for a local problem, i.e., the hyperbolic boundary should satisfy a particular condition near the parabolic points and the elliptic boundary should be close to the parabolic line. In order to apply these multipliers to a practical problem, an admissible domain was modified in a δ-neighborhood of the parabolic points. This is called a geometric regularization. By using compact operator theory, the solution is extended further into the general elliptic domain. Note that the local geometry converges to the original geometry as δ approaches to zero. Hence, it is conjectured that the nature of the singularity is retained, and in the limit is the same order as the fundamental solution...