Javier de Frutos

A Spectral Element Method for the Navier--Stokes Equations with Improved Accuracy

We present a procedure to enhance the accuracy of spectral element methods for the evolutionary Navier--Stokes equations. The approximations to the velocity and the pressure obtained using a spectral element method are postprocessed by solving a discrete Stokes problem only once, when the time evolution is completed. The analysis shows that this procedure improves the order of convergence for the approximations to the velocity and the pressure. The choice of a discrete pressure space endowed with an inf-sup condition independent of the discretization parameter leads to optimal error bounds for the pressure.

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

Summary. We present an approximate-inertial-manifold-based postprocess to enhance Chebyshev or Legendre spectral Galerkin methods. We prove that the postprocess improves the order of convergence of the Galerkin solution, yielding the same accuracy as the nonlinear Galerkin method. Numerical experiments show that the new method is computationally more efficient than Galerkin and nonlinear Galerkin methods. New approximation results for Chebyshev polynomials are presented.

A spectral method for bonds

We present an spectral numerical method for the numerical valuation of bonds with embedded options. We use a CIR model for the short-term interest rate. The method is based on a Galerkin formulation of the partial differential equation for the value of the bond, discretized by means of orthogonal Laguerre polynomials. The method is shown to be very efficient, with a high precision for the type of problems treated here and is easy to use with more general models with nonconstant coefficients. As a consequence, it can be a possible alternative to other approaches employed in practice, specially when a calibration of the parameters of the model is needed to match the observed market data.

A postprocess based improvement of the spectral element method

Abstract We present an approximate inertial manifold based postprocess to enhance the accuracy of the spectral element method for evolutionary equations of dissipative type. The analysis shows that the postprocess improves the order of convergence. The postprocess consists on the resolution of a discrete elliptic problem only once when the time evolution has been completed. The overcost is then negligible and the resulting postprocessed method is really an improvement over the standard spectral element discretization to which is applied.

Postprocessing the Linear Finite Element Method

We extend the idea of the postprocessing Galerkin methods for dissipative evolution equations to the case of the linear finite element method. The postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods. The analysis shows that this procedure improves the order of convergence of the piecewise linear Galerkin finite element approximation in the H1 norm. We show by means of numerical experiments that there is no improvement in the order of convergence in the L2 norm.

The Postprocessed Mixed Finite-Element Method for the Navier-Stokes Equations: Refined Error Bounds

A postprocessing technique for mixed finite-element methods for the incompressible Navier-Stokes equations is analyzed. The postprocess, which amounts to solving a (linear) Stokes problem, is shown to increase the order of convergence of the method to which it is applied by one unit (times the logarithm of the mesh diameter). In proving the error bounds, some superconvergence results are also obtained. Contrary to previous analysis of the postprocessing technique, in the present paper we take into account the loss of regularity suffered by the solutions of the Navier-Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data. As in [H. G. Heywood and R. Rannacher, SIAM J. Numer. Anal., 25 (1988), pp. 489-512], where the same hypothesis is assumed, no better than fifth-order convergence is achieved.

Optimal error bounds for two-grid schemes applied to the Navier-Stokes equations

Abstract We consider two-grid mixed-finite element schemes for the spatial discretization of the incompressible Navier–Stokes equations. A standard mixed-finite element method is applied over the coarse grid to approximate the nonlinear Navier–Stokes equations while a linear evolutionary problem is solved over the fine grid. The previously computed Galerkin approximation to the velocity is used to linearize the convective term. For the analysis we take into account the lack of regularity of the solutions of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data. Optimal error bounds are obtained.

A posteriori error estimation with the p-version of the finite element method for nonlinear parabolic differential equations

We analyze an a posteriori error estimator for nonlinear parabolic differential equations in several space dimensions. The spatial discretization is carried out using the p-version of the finite element method. The error estimates are obtained by solving an elliptic problem at the desired times when the estimation is wanted. Some numerical experiments prove the efficiency of the error estimation.

Grad-div Stabilization for the Evolutionary Oseen Problem with Inf-sup Stable Finite Elements

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank–Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results.

Implicit–explicit Runge–Kutta methods for financial derivatives pricing models

Abstract Implicit–explicit Runge–Kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach. The methods are showed to be an alternative to other existing procedures for the numerical valuation of American type contracts. We follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements, finite differences and finite volume methods. Some numerical experiments are presented.