Stability Estimates for ℎ-𝑝 Spectral Element Methods for Elliptic Problems

Proceedings Mathematical Sciences

Satyendra Tomar³  Pravir Dutt²  B V Rathish Kumar¹ 

[1] $$;Department of Mathematics, Indian Institute of Technology, Kanpur 0 0, India$$;Post Doctoral Researcher, Department of Mathematical Physics and Computational Mechanics, University of Twente, The Netherland$$

关键词: Corner singularities;    geometrical mesh;    modified polar coordinates;    quasi-uniform mesh;    fractional Sobolev norms;    stability estimate;    polylogarithmic bounds.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $r_k$, where $r_k$ measures the distance between the point 𝑃 and the vertex $A_k$ in a sectoral neighborhood of each of these vertices. In each of these sectoral neighborhoods we use a local coordinate system $(𝜏_k, 𝜃_k)$ where $𝜏_k = ln r_k$ and $(r_k, 𝜃_k)$ are polar coordinates with origin at $A_k$, 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.

【 授权许可】

Unknown