For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been ... more For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been very successfully used in practical problems. However, in many engineering and scientific applications we frequently encounter the numerical solutions of elliptic boundary value problems in non-smooth domains which give rise to singularities in the solution. In such cases the accuracy of the solution obtained by SEM deteriorates
We propose a new parallel h-p Spectral element method to solve elliptic boundary value problems w... more We propose a new parallel h-p Spectral element method to solve elliptic boundary value problems with mixed Neumann and Dirichlet boundary conditions on non-smooth domains. The method is shown to be exponentially accurate and asymptotically faster than the stan-dard h-p finite element method. We use the auxiliary mapping of the form of z = ln ξ. The spectral element functions we use are fully non-conforming for pure Dirichlet prob-lems. However, for mixed problems we need to make it conforming only at the vertices of the elements. The dimension of the resulting Schur complement matrix is quite small and this enables us to construct an accurate approximation of the Schur system. The method is a least-squares collocation method and the resulting normal equations are solved using preconditioned conjugate gradient method. We state the differentiability, stability and error estimates and discuss the numerical scheme and parallelization techniques. Using the sta-bility estimates a parallel...
The problem of estimating spatially varying coefficients of partial differential equations is con... more The problem of estimating spatially varying coefficients of partial differential equations is considered from observation of the solution and of the right hand side of the equation. It is assumed that the observations are distributed in the domain and that enough observations are given. A method of discretization and an efficient multigrid method for solving the resulting discrete systems are described. Numerical results are presented for estimation of coefficients in an elliptic and a parabolic partial differential equation.
It is well known that elliptic problems when posed on non-smooth domains, develop singularities. ... more It is well known that elliptic problems when posed on non-smooth domains, develop singularities. We examine such problems within the framework of spectral element methods and resolve the singularities with exponential accuracy.
... SK Tomar1, P. Dutt2, and BV Rathish Kumar2 1 Faculty of Mathematical Sciences, University of ... more ... SK Tomar1, P. Dutt2, and BV Rathish Kumar2 1 Faculty of Mathematical Sciences, University of Twente PO Box 217, 7500 AE, Enschede, The Netherlands [email protected] 2 Department of Mathematics IIT Kanpur, 208016, UP, India {pravir,bvrk}@iitk.ac.in Abstract. ...
ABSTRACT Several methods have been proposed in the literature for solving the Black–Scholes equat... more ABSTRACT 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.
In this paper we show that we can use a modified version of the h-p spectral element method propo... more In this paper we show that we can use a modified version of the h-p spectral element method proposed in 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 . 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 and the h-p finite element method and stronger error estimates are obtained.
In this paper we show that the h-p spectral element method developed in applies to elliptic probl... more In this paper we show that the h-p spectral element method developed in 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.
In this paper a method is developed for solving hyperbolic initial boundary value problems in one... more In this paper a method is developed for solving hyperbolic initial boundary value problems in one space dimension using domain decomposition, which can be extended to problems in several space dimensions. We minimize a functional which is the sum of squares of the L 2 norms of the residuals and a term which is the sum of the squares of the L 2 norms of the jumps in the function across interdomain boundaries. To make the problem well posed the interdomain boundaries are made to move back and forth at alternate time steps with sufficiently high speed. We construct parallel preconditioners and obtain error estimates for the method.
ABSTRACT In this paper a nonconforming spectral element method is discussed for the elliptic part... more ABSTRACT In this paper a nonconforming spectral element method is discussed for the elliptic partial differential equations with periodic boundary conditions. The formulation is based on the minimization of a functional by the least squares method. The periodic boundary conditions are added in the weak form in the formulation of the functional and the normal structure of resulting coefficient matrix is retained. To obtain the conforming solution a set of corrections are made and the error is estimated in H1H1 norm.
Spectral methods give very highly accurate approximations to hyperbolic problems with smooth solu... more Spectral methods give very highly accurate approximations to hyperbolic problems with smooth solutions. The naive use of spectral methods on hy-perbolic problems with discontinuous solutions, however, produces oscillatory numerical results. The oscillations arising ...
In this paper we present a modified version of the pseudospectral method for solving initial-boun... more In this paper we present a modified version of the pseudospectral method for solving initial-boundary value systems of hyperbolic partial differential equations. We are able to avoid problems of instability by regularizing the boundary conditions. We prove the stability and convergence of our proposed scheme and obtain error estimates.
ABSTRACT We propose a nonconforming spectral/hp element method for solving elliptic systems on no... more 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.
Journal of Computational and Applied Mathematics, 2001
... Spectral methods for hyperbolic initial boundary value problems on parallel computers *1. Pra... more ... Spectral methods for hyperbolic initial boundary value problems on parallel computers *1. Pravir K. Dutt Corresponding Author Contact Information , a and Smita Bedekar b. a Department of Mathematics, Indian Institute of Technology, Kanpur 208016, India. ...
Journal of Computational and Applied Mathematics, 2008
In this paper we propose preconditioners for spectral element methods for elliptic and parabolic ... more In this paper we propose preconditioners for spectral element methods for elliptic and parabolic problems. These preconditioners are constructed using separation of variables and are easy to invert. Moreover they are spectrally equivalent to the quadratic forms which they are used to approximate.
Journal of Computational and Applied Mathematics, 2007
A spectral element method for solving parabolic initial boundary value problems on smooth domains... more A spectral element method for solving parabolic initial boundary value problems on smooth domains using parallel computers is presented in this paper. The space domain is divided into a number of shape regular quadrilaterals of size h and the time step k is proportional to h 2 . At each time step we minimize a functional which is the sum of the squares of the residuals in the partial differential equation, initial condition and boundary condition in different Sobolev norms and a term which measures the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The Sobolev spaces used are of different orders in space and time. We can define a preconditioner for the minimization problem which allows the problem to decouple. Error estimates are obtained for both the h and p versions of this method.
For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been ... more For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been very successfully used in practical problems. However, in many engineering and scientific applications we frequently encounter the numerical solutions of elliptic boundary value problems in non-smooth domains which give rise to singularities in the solution. In such cases the accuracy of the solution obtained by SEM deteriorates
We propose a new parallel h-p Spectral element method to solve elliptic boundary value problems w... more We propose a new parallel h-p Spectral element method to solve elliptic boundary value problems with mixed Neumann and Dirichlet boundary conditions on non-smooth domains. The method is shown to be exponentially accurate and asymptotically faster than the stan-dard h-p finite element method. We use the auxiliary mapping of the form of z = ln ξ. The spectral element functions we use are fully non-conforming for pure Dirichlet prob-lems. However, for mixed problems we need to make it conforming only at the vertices of the elements. The dimension of the resulting Schur complement matrix is quite small and this enables us to construct an accurate approximation of the Schur system. The method is a least-squares collocation method and the resulting normal equations are solved using preconditioned conjugate gradient method. We state the differentiability, stability and error estimates and discuss the numerical scheme and parallelization techniques. Using the sta-bility estimates a parallel...
The problem of estimating spatially varying coefficients of partial differential equations is con... more The problem of estimating spatially varying coefficients of partial differential equations is considered from observation of the solution and of the right hand side of the equation. It is assumed that the observations are distributed in the domain and that enough observations are given. A method of discretization and an efficient multigrid method for solving the resulting discrete systems are described. Numerical results are presented for estimation of coefficients in an elliptic and a parabolic partial differential equation.
It is well known that elliptic problems when posed on non-smooth domains, develop singularities. ... more It is well known that elliptic problems when posed on non-smooth domains, develop singularities. We examine such problems within the framework of spectral element methods and resolve the singularities with exponential accuracy.
... SK Tomar1, P. Dutt2, and BV Rathish Kumar2 1 Faculty of Mathematical Sciences, University of ... more ... SK Tomar1, P. Dutt2, and BV Rathish Kumar2 1 Faculty of Mathematical Sciences, University of Twente PO Box 217, 7500 AE, Enschede, The Netherlands [email protected] 2 Department of Mathematics IIT Kanpur, 208016, UP, India {pravir,bvrk}@iitk.ac.in Abstract. ...
ABSTRACT Several methods have been proposed in the literature for solving the Black–Scholes equat... more ABSTRACT 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.
In this paper we show that we can use a modified version of the h-p spectral element method propo... more In this paper we show that we can use a modified version of the h-p spectral element method proposed in 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 . 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 and the h-p finite element method and stronger error estimates are obtained.
In this paper we show that the h-p spectral element method developed in applies to elliptic probl... more In this paper we show that the h-p spectral element method developed in 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.
In this paper a method is developed for solving hyperbolic initial boundary value problems in one... more In this paper a method is developed for solving hyperbolic initial boundary value problems in one space dimension using domain decomposition, which can be extended to problems in several space dimensions. We minimize a functional which is the sum of squares of the L 2 norms of the residuals and a term which is the sum of the squares of the L 2 norms of the jumps in the function across interdomain boundaries. To make the problem well posed the interdomain boundaries are made to move back and forth at alternate time steps with sufficiently high speed. We construct parallel preconditioners and obtain error estimates for the method.
ABSTRACT In this paper a nonconforming spectral element method is discussed for the elliptic part... more ABSTRACT In this paper a nonconforming spectral element method is discussed for the elliptic partial differential equations with periodic boundary conditions. The formulation is based on the minimization of a functional by the least squares method. The periodic boundary conditions are added in the weak form in the formulation of the functional and the normal structure of resulting coefficient matrix is retained. To obtain the conforming solution a set of corrections are made and the error is estimated in H1H1 norm.
Spectral methods give very highly accurate approximations to hyperbolic problems with smooth solu... more Spectral methods give very highly accurate approximations to hyperbolic problems with smooth solutions. The naive use of spectral methods on hy-perbolic problems with discontinuous solutions, however, produces oscillatory numerical results. The oscillations arising ...
In this paper we present a modified version of the pseudospectral method for solving initial-boun... more In this paper we present a modified version of the pseudospectral method for solving initial-boundary value systems of hyperbolic partial differential equations. We are able to avoid problems of instability by regularizing the boundary conditions. We prove the stability and convergence of our proposed scheme and obtain error estimates.
ABSTRACT We propose a nonconforming spectral/hp element method for solving elliptic systems on no... more 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.
Journal of Computational and Applied Mathematics, 2001
... Spectral methods for hyperbolic initial boundary value problems on parallel computers *1. Pra... more ... Spectral methods for hyperbolic initial boundary value problems on parallel computers *1. Pravir K. Dutt Corresponding Author Contact Information , a and Smita Bedekar b. a Department of Mathematics, Indian Institute of Technology, Kanpur 208016, India. ...
Journal of Computational and Applied Mathematics, 2008
In this paper we propose preconditioners for spectral element methods for elliptic and parabolic ... more In this paper we propose preconditioners for spectral element methods for elliptic and parabolic problems. These preconditioners are constructed using separation of variables and are easy to invert. Moreover they are spectrally equivalent to the quadratic forms which they are used to approximate.
Journal of Computational and Applied Mathematics, 2007
A spectral element method for solving parabolic initial boundary value problems on smooth domains... more A spectral element method for solving parabolic initial boundary value problems on smooth domains using parallel computers is presented in this paper. The space domain is divided into a number of shape regular quadrilaterals of size h and the time step k is proportional to h 2 . At each time step we minimize a functional which is the sum of the squares of the residuals in the partial differential equation, initial condition and boundary condition in different Sobolev norms and a term which measures the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The Sobolev spaces used are of different orders in space and time. We can define a preconditioner for the minimization problem which allows the problem to decouple. Error estimates are obtained for both the h and p versions of this method.
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