Stability estimates for h-p spectral element methods for elliptic problems

Abstract

In a series of papers of which this is the first we study how to solve elliptic problems on polygonal domains using spectral methods on parallel computers. To overcome the singularities that arise in a neighborhood of the corners we use a geometrical mesh. With this mesh we seek a solution which minimizes 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 by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in an appropriate fractional Sobolev norm, to the functional being minimized. Since the second derivatives of the actual solution are not square integrable in a neighborhood of the corners we have to multiply the residuals in the partial differential equation by an appropriate power of rk , where rk measures the distance between the point P and the vertex Ak in a sectoral neighborhood of each of these vertices. In each of these sectoral neighborhoods we use a local coordinate system .k; k/ where k D ln rk and .rk; k/ are polar coordinates with origin at Ak , as first proposed by Kondratiev. We then derive differentiability estimates with respect to these new variables and a stability estimate for the functional we minimize. In [6] we will show that we can use the stability estimate to obtain parallel preconditioners and error estimates for the solution of the minimization problem which are nearly optimal as the condition number of the preconditioned system is polylogarithmic in N, the number of processors and the number of degrees of freedom in each variable on each element. Moreover if the data is analytic then the error is exponentially small in N.