We propose a nonconforming spectral/hp element method for solving elliptic systems on non smooth domains using parallel computers. A geometric mesh is used in a neighbourhood of the corners and a modified set of polar coordinates, as defined by Kondratiev [Diff. Equations 6: 1392 – 1401, 1970], is introduced in these neighbourhoods. In the remaining part of the domain Cartesian coordinates are used. 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. The set of common boundary values consists only of the values of the spectral element functions at the vertices of the polygonal domain. Since the cardinality of the set of common boundary values is so small, a nearly exact Schur complement matrix can be computed. The method is exponentially accurate and asymptotically faster than the h-p finite element method. The normal equations obtained from the least-squares formulation can be solved by the preconditioned conjugate gradient method using a parallel preconditioner. The algorithm is implemented on a distributed memory parallel computer with small inter- processor communication. Numerical results for scalar problems and the equations of elasticity are provided to validate the error estimates and estimates of computational complexity that have been obtained.
Exploiting the theory of fractional Fourier transform, the wavelet convolution product and existence theorems associated with the n-dimensional wavelet transform are investigated and their properties studied.
The infinite pseudo-differential operator on WM(Rn) space is introduced and its various properties are studied. A general class of symbols θ(x,ξ) is introduced and then it is proved that the pseudo-differential operator Aθφ is a continuous linear mapping from WM(Rn) into itself. An Lp(Rn)-boundedness result for the pseudo-differential operator associated with a general class of symbols σ(x,ξ) for ξ = u+it is obtained. It is shown that the pseudo-differential operator is a bounded linear operator from Lp(Rn) into Lp(Rn) for 1 < p < ∞. The Sobolev space of type Gs,p(Rn) is introduced and its properties are studied.
In this paper the space G2(ℝ) is introduced and the boundedness, normality and spectral properties of the Hausdorff operator Hϕ; acting on the space G2(ℝ) is investigated by using the theory of Fourier transform and Hilbert transform. Existence of spectrum of the Hausdorff operator on G2(ℝ) is obtained.
In this study, we developed the two-dimensional Legendre wavelet modified Petrov–Galerkin method for solving the two-dimensional moving boundary problem arising during melting of solid whose one surface is kept under most generalised boundary condition, and other two surfaces are insulated. The particular cases when surface subjected to the boundary condition of first, second and third kinds are discussed in detail. For validity of the present method, we have plotted graphs between residual (obtained from the original differential equation and its associated boundary conditions) and x-axis and found the effect of an error on moving layer thickness and y coordinate, respectively. Furthermore, we proved the convergence analysis of present method. The effect of parameters (Predvoditelev number, Kirpichev number, Biot number) on the moving layer thickness is discussed in detail. The whole analysis is presented in a dimensionless form.