A Difference Method for Boundary Value Problems of the Third Kind

Pure difference methods for elliptic boundary value problems with derivative boundary conditions are treated by Batschelet [1], Giese [2], Lebedev [3-8], Volkov [9-10] and Wigley [11], etc. For the same problems a kind of difference methods, what is called "Finite-element-method", are also investigated by Demjanovic [12], Friedrichs and Keller [IS], Oganesyan [14-15], Oganesyan and Rukovetz [16-171 and Zlamal [18-19], etc. In this method a reduced minimal problem from the original boundary value problem is solved approximately in a subspace spanned by a class of finite number of "element" functions and their translated functions. The resulting difference scheme approximates automatically the differential equation in the interior of the domain and the boundary condition at points near the boundary. In these works the estimate of error between the exact and approximate solutions is given either in order of mesh width or precisely in an explicit form. On the other hand, as far as we know, there were few works about difference methods for hyperbolic and parabolic mixed initial and boundary value problems with derivative boundary conditions in a domain of any shape. From mathematical interest we can refer to Lions [20] and Chekhlov [21] whose method is called "penalty method", in which the