The present article is the first one of the trilogy of papers [see the following entries (Zbl 0723.73075; Zbl 0723.73076)] on the development of an adaptive h-p version (h: sizes of the elements; p: orders of the shape functions) of the finite element method for the solution of various boundary value problems in solid and fluid mechanics. For the h-p method represents significant departures from conventional finite element techniques, a resolution of several formidable problems in their effective implementation is required, such as new data structures, equation solvers and certain criteria for choosing a distribution of mesh sizes and approximate orders.
In Part 1 of the presentation, general formulation issues are taken up, the data structure is developed, and the h-p adaptive strategy is introduced, thus contributing to a finite element scheme for linear elliptic boundary-value problems characterized by general elliptic systems of partial differential equations. In particular, the h- and p- adaptive is discussed, restricting the irregularity of the considered meshes to the index one. Furthermore, the concept of constrained approximation is outlined and its impact on such basic ingredients to the FEM as element stiffness matrix and load vector calculations is presented. The following discussion on the h-p adaptive finite element method is then restricted to the two-dimensional case, where the initial mesh is topologically a portion of a regular, rectangular grid in \({\mathbb{R}}^ 2\). Finally, some details concerning the data structure are given. The paper concludes with a presentation of four illustrative examples and some details on forms of the corresponding variational formulations.