• C S Upadhyay

      Articles written in Proceedings – Mathematical Sciences

    • Nonconforming $h-p$ Spectral Element Methods for Elliptic Problems

      P K Dutt N Kishore Kumar C S Upadhyay

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      In this paper we show that we can use a modified version of the $h-p$ spectral element method proposed in [6,7,13,14] to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal domains using nonconforming spectral element functions. A geometrical mesh is used in a neighbourhood of the corners. With this mesh we seek a solution which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized. In the neighbourhood of the corners, modified polar coordinates are used and a global coordinate system elsewhere. A stability estimate is derived for the functional which is minimized based on the regularity estimate in [2]. 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.

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

      P Dutt Akhlaq Husain A S Vasudeva Murthy C S Upadhyay

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      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

      P Dutt Akhlaq Husain A S Vasudeva Murthy C S Upadhyay

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      This is the second of a series of papers devoted to the study of $h-p$ spectral element methods for three dimensional elliptic problems on non-smooth domains. The present paper addresses the proof of the main stability theorem.We assume that the differential operator is a strongly elliptic operator which satisfies Lax–Milgram conditions. The spectral element functions are non-conforming. The stability estimate theorem of this paper will be used to design a numerical scheme which give exponentially accurate solutions to three dimensional elliptic problems on non-smooth domains and can be easily implemented on parallel computers.

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