h-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-I: Regularity estimates and stability theorem

Abstract

This is the first of a series of papers devoted to the study of h-p spectral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the vertex-edge singularities. In addition, the solution is anisotropic in the neighbourhoods of the edges and vertex-edges. To overcome the singularities which arise in the neighbourhoods of vertices, vertex-edges and edges, we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. The geometrical mesh becomes a quasi-uniform mesh in the new system of coordinates. We then derive differentiability estimates in these new set of variables and state our main stability estimate theorem using a non-conforming h-p spectral element method whose proof is given in a separate paper.

h- p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-II: Proof of stability theorem

“…The data F , g [0] and g [1] are analytic on each open face and g [0] is continuous on i∈D¯ i . In [5,13], we divided the domain into a regular region, a set of vertex neighbourhoods, a set of edge neighbourhoods and a set of vertex-edge neighbourhoods. A set of modified coordinates have been defined in these neighoubrhoods which are modifications of the standard spherical and cylindrical coordinates.…”

“…S v is divided into triangular and quadrilateral elements S v j for 1 ≤ j ≤ I v , where I v is a fixed constant. We impose a geometric mesh in the vertex neighbourhood v of the vertex v as shown in figure 3(b) (see [5,13] for details).…”

“…(3.44) Moreover, from (3.33) A geometric mesh is imposed on v−e as shown in figure 4 (see [5,13] for details).…”

“…This paper is the second of a series of papers devoted to the study of the h-p spectral element method for solving three dimensional elliptic problems on non-smooth domains using parallel computers. The first paper [5], which deals with the regularity of the solution in the neighbourhoods of vertices, edges and vertex-edges and description of the main stability theorem, will be published separately. The present paper develops the proof of the basic stability estimate of [5] based on the work in [13].…”

“…The first paper [5], which deals with the regularity of the solution in the neighbourhoods of vertices, edges and vertex-edges and description of the main stability theorem, will be published separately. The present paper develops the proof of the basic stability estimate of [5] based on the work in [13]. In the forthcoming work, we shall provide the numerical scheme, the parallel preconditioner, error estimates and the solution techniques based on the stability estimates.…”