Erratum to ``A C1-P2 finite element without nodal basis''

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A quadratic finite element wavelet Riesz basis

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in [Formula: see text]. The wavelets are stable in [Formula: see text] for [Formula: see text] and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for [Formula: see text] are provided for the unit square.

Spectral Method with the Tensor-Product Nodal Basis for the Steklov Eigenvalue Problem

This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Steklov eigenvalue problem. A priori error estimates of spectral method are discussed, and based on the work of Melenk and Wohlmuth (2001), a posterior error estimator of the residual type is given and analyzed. In addition, this paper combines the shifted-inverse iterative method and spectral method to establish an efficient scheme. Finally, numerical experiments with MATLAB program are reported.

A Unstructured Nodal Spectral-Element Method for the Navier-Stokes Equations

An unstructured nodal spectral-element method for the Navier-Stokes equations is developed in this paper. The method is based on a triangular and tetrahedral rational approximation and an easy-to-implement nodal basis which fully enjoys the tensorial product property. It allows arbitrary triangular and tetrahedral mesh, affording greater flexibility in handling complex domains while maintaining all essential features of the usual spectral-element method. The details of the implementation and some numerical examples are provided to validate the efficiency and flexibility of the proposed method.

WAVELET OPTIMIZED VALUATION OF FINANCIAL DERIVATIVES

We introduce a simple but efficient PDE method that makes use of interpolation wavelets for their advantages in compression and interpolation in order to define a sparse computational domain. It uses finite difference filters for approximate differentiation, which provide us with a simple and sparse stiffness matrix for the discrete system. Since the method only uses a nodal basis, the application of non-constant terms, boundary conditions and free-boundary conditions is straightforward. We give empirical results for financial products from the equity and fixed income markets in 1, 2 and 3 dimensions and show a speed-up factor between 2 and 4 with no significant reduction of precision.