Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains

Abstract

Abstract. In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.

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

“…The proof is similar to the proof of Lemmas 3.1 and 3.4 in [7]. We define the vector field w by w = A∇ x u.…”

Nonconforming h-p spectral element methods for elliptic problems

“…We examine how to parallelize the method and show that the set of common boundary values consists of the values of the function at the corners of the polygonal domain. The method is faster than that proposed in [6,7,14] and the h-p finite element method and stronger error estimates are obtained. …”

“…For the modified version of the h-p spectral element method examined here a stability estimate is proved which is based on the regularity estimate of Babuska and Guo in [2]. The proof is much simpler than that of the stability estimate in [6,7]. Moreover the error estimates are stronger.…”

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

“…We now seek a solution to elliptic BVP's as in [18,19,[30][31][32] which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and a fractional Sobolev norm of the residuals in the boundary conditions and enforce continuity across inter element boundaries by adding a term which measures the sum of the squares of the jump in the function and its derivatives at inter element boundaries in appropriate Sobolev norms to the functional being minimized. Here we examine the non-conforming version of the method.…”

“…In [18,19], an exponentially accurate h-p spectral element method was proposed for two dimensional elliptic problems on non-smooth domains with analytic coefficients posed on curvilinear polygons with piecewise analytic boundary. The method is able to resolve the singularities which arise at the corners using a geometrical mesh as proposed by Babuška and Guo. In contrast to the two dimensional case, the character of the singularities in three dimensions is much more complex not only because of higher dimension but also due to the varied nature of the singularities which are the vertex singularity, the edge singularity and the vertex-edge singularity.…”