ABSTRACT
Fourth-order elliptic boundary value problems can be reduced to operator equations in Hilbert spaces G that are certain subspaces of the Sobolev space. Construction of asymptotically optimal grid approximations and, most particularly, asymptotically optimal algorithms are very difficult because the associated spline subspaces are not of Lagrangian type. These difficulties evoked a series of attempts to reduce such problems to second-order differential equations, but with no essential progress in the construction of asymptotically optimal algorithms. This chapter discusses the new aspects connected with the study of PGMs and iterative algorithms in the presence of linear constraints. More traditional questions, deeding with such issues as the use of quadrature formulas, curvilinear triangles, and piecewise polynomial and singular basis functions can be considered by analogy to investigations carried out for standard elliptic boundary value problems.
