We develop and test spectral Galerkin schemes to solve the coupled Orr–Sommerfeld (OS) and induction equationsfor parallel, incompressible MHD in free-surface and fixed-boundary geometries. The schemes’ discrete bases consistof Legendre internal shape functions, supplemented with nodal shape functions for the weak imposition of thestress and insulating boundary conditions. The o...

We address quadrature-based approximations of the bilinear inverse form u>A−1u, where A is a real symmetric positive definite matrix, and analyze properties of the Gauss, Gauss-Radau, and Gauss-Lobatto quadrature. In particular, we establish monotonicity of the bounds given by these quadrature rules, compare the tightness of these bounds, and derive associated convergence rates. To our knowledg...

This paper focuses on increasing the accuracy of low-order four-node quadrilateral finite elements for the transient analysis of wave propagation. Modified integration rules, originally proposed for time-harmonic problems, provide the basis for the proposed technique. The modified integration rules shift the integration points to locations away from the conventional Gauss or Gauss-Lobatto integ...

Properties of the hybrid of block-pulse functions and Lagrange polynomials based on the Legendre-Gauss-type points are investigated and utilized to define the composite interpolation operator as an extension of the well-known Legendre interpolation operator. The uniqueness and interpolating properties are discussed and the corresponding differentiation matrix is also introduced. The appl...

Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several ...

properties of the hybrid of block-pulse functions and lagrange polynomials based on the legendre-gauss-type points are investigated and utilized to define the composite interpolation operator as an extension of the well-known legendre interpolation operator. the uniqueness and interpolating properties are discussed and the corresponding differentiation matrix is also introduced. the applicabili...

This paper deals with Vandermonde matrices Vn whose nodes are the Gauss–Lobatto Chebyshev nodes, also called extrema Chebyshev nodes. We give an analytic factorization and explicit formula for the entries of their inverse, and explore its computational issues. We also give asymptotic estimates of the Frobenius norm of both Vn and its inverse and present an explicit formula for the determinant o...

The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation n...