A. S. Vasudeva Murthy

h - p Spectral element methods for three dimensional elliptic problems on non-smooth domains

Elliptic partial differential equations arise in many fields of science and engineering such as steady state distribution of heat, fluid dynamics, structural/mechanical engineering, aerospace engineering and seismology etc. In three dimensions it is well known that the solutions of elliptic boundary value problems have singular behavior near the corners and edges of the domain. The singularities which arise are known as vertex, edge, and vertex-edge singularities. We propose a nonconforming h-p spectral element method to solve three dimensional elliptic boundary value problems on non-smooth domains to exponential accuracy. To overcome the singularities which arise in the neighbourhoods of the 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. We then derive differentiability estimates in these new set of variables and a stability estimate on which our method is based for a non-conforming h-p spectral element method. The Sobolev spaces in vertex-edge and edge neighbourhoods are anisotropic and become singular at the corners and edges. The method is essentially a least-squares collocation} method and a solution can be obtained using Preconditioned Conjugate Gradient Method (PCGM). To solve the minimization problem we need to solve the normal equations for the least-squares problem. The residuals in the normal equations can be obtained without computing and storing mass and stiffness matrices. Computational results for a number of model problems confirm the theoretical estimates obtained for the error and computational complexity.

A novel variable resolution global spectral method on the sphere

Highlights? HTBT is a re-parametrization map on the sphere, preserves differential operators. ? Variable resolution global spectral method is generated through re-parametrization. ? Resolution is finer over the tropics and decreases smoothly away from it. ? Spectral coefficients are computed through FFT and Gaussian quadrature methods. ? Boyd-Vandeven filter helps to retain the higher order accuracy of the method. A variable resolution global spectral method is created on the sphere using High resolution Tropical Belt Transformation (HTBT). HTBT belongs to a class of map called reparametrisation maps. HTBT parametrisation of the sphere generates a clustering of points in the entire tropical belt; the density of the grid point distribution decreases smoothly in the domain outside the tropics. This variable resolution method creates finer resolution in the tropics and coarser resolution at the poles. The use of FFT procedure and Gaussian quadrature for the spectral computations retains the numerical efficiency available with the standard global spectral method. Accuracy of the method for meteorological computations are demonstrated by solving Helmholtz equation and non-divergent barotropic vorticity equation on the sphere.

On the Wave Equations of Kirchhoff–Narasimha and Carrier

A nonlinear nonlocal wave equation modelling the coupling between transverse and longitudinal vibrations was derived by Carrier in 1945. In 1968, using careful asymptotics, Narasimha derived a similar equation but with a different nonlinearity (nowadays referred as Kirchhoff type nonlinearity). In this study we solve both the equations numerically and compare them. Since there are no experimental data available it is not possible to suggest which is the better model. However Kurmyshev has pointed out that Carrier’s model cannot be valid for rubber or soft nylon strings.