Pravir Dutt

Stable boundary conditions and difference schemas for Navier-Stokes equations

The Navier–Stokes equations at high Reynolds numbers can be viewed as an incompletely elliptic perturbation of the Euler equations. By using the entropy function for the Euler equations as a measure of “energy” for the Navier–Stokes equations we are able to obtain nonlinear “energy” estimates for the mixed initial boundary value problem. These estimates are used to derive boundary conditions that guarantee

Optimal time advancing dispersion relation preserving schemes

L^2

Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains

boundedness even when the Reynolds number tends to infinity. Finally, we propose a new difference scheme for modeling the Navier–Stokes equations in multidimensions for which we are able to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation.

Spectral methods for hyperbolic initial boundary value problems on parallel computers

In this paper we examine the constrained optimization of explicit Runge-Kutta (RK) schemes coupled with central spatial discretization schemes to solve the one-dimensional convection equation. The constraints are defined with respect to the correct error propagation equation which goes beyond the traditional von Neumann analysis developed in Sengupta et al. [T.K. Sengupta, A. Dipankar, P. Sagaut, Error dynamics: beyond von Neumann analysis, J. Comput. Phys. 226 (2007) 1211-1218]. The efficiency of these optimal schemes is demonstrated for the one-dimensional convection problem and also by solving the Navier-Stokes equations for a two-dimensional lid-driven cavity (LDC) problem. For the LDC problem, results for Re=1000 are compared with results using spectral methods in Botella and Peyret [O. Botella, R. Peyret, Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids 27 (1998) 421-433] to calibrate the method in solving the steady state problem. We also report the results of the same flow at Re=10,000 and compare them with some recent results to establish the correctness and accuracy of the scheme for solving unsteady flow problems. Finally, we also compare our results for a wave-packet propagation problem with another method developed for computational aeroacoustics.

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

In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.

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

Abstract In this paper a parallel spectral algorithm is developed for hyperbolic initial boundary value problems in one space dimension. The Galerkin-Collocation method, which is spectrally accurate in both space and time, is parallelized by using domain decomposition. This procedure leads to a minimization problem in which there is coupling at inter-domain boundaries. We construct a decoupled preconditioner which can be used to iteratively solve the minimization problem. Symmetric formulation of the problem, which is needed to compute the residual for the normal equations, is discussed. The methodology outlined for computing the normal equations applies equally well to computation of the residual for the p and h–p versions of the finite element method. There is, therefore, no need to compute the mass and stiffness matrices to obtain the residual, as is normally done. This leads to a great saving in time and memory particularly for solving nonlinear problems using the p and h–p versions of the finite element method. The method we discuss in this paper generalizes to hyperbolic initial boundary value problems in multidimensions too, provided the computational boundaries we have introduced are noncharacteristic and the system is symmetrizable. Finally, we show that for the case of analytic coefficients and data, satisfying all the required compatibility conditions so that the solution is analytic, the numerical solution is exponentially accurate in N, where N is proportional to the number of subdomains and the number of degrees of freedom in each element.

Nonconforming spectral/hp element methods for elliptic systems

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 pointP and the vertexAk in a sectoral neighborhood of each of these vertices. In each of these sectoral neighborhoods we use a local coordinate system (τk, θk) where τk= lnrk and (rk, θk) 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 inN, 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 inN.

Nonconforming Least-Squares Spectral Element Method for European Options

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.

Spectral methods for periodic initial value problems with nonsmooth data

Abstract 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.

A spline-based parameter estimation technique for static models of elastic structures

Several methods have been proposed in the literature for solving the Black-Scholes equation for European Options. The method proposed in the current study achieves spectral accuracy in both space and time. The method is based on minimization of a functional given in terms of the sum of squares of the residuals in the partial differential equation and initial condition in different Sobolev norms, and a term which measures the jump in the function and its derivatives across inter-element boundaries in appropriate fractional Sobolev norms. To obtain values of the solution and its derivatives the initial condition is mollified and the computed solution is post processed. Error estimates are obtained for this method. Specific numerical examples are given to show the efficiency of this method.