On polynomials orthogonal with respect to certain Sobolev inner products

AbstractWe are concerned with polynomials {pn(λ)} that are orthogonal with respect to the Sobolev inner product 〈 f, g 〉λ = ∝ fg dϑ + λ ∝ f′g′ dψ, where λ is a non-negative constant. We show that if the Borel measures dϑ and dψ obey a specific condition then the Pn(λ)'s can be expanded in the polynomials orthogonal with respect to dϑ in such a manner that, subject to correct normalization, the expansion coefficients, except for the last, are independent of n and are themselves orthogonal polynomials in λ. We explore several examples and demonstrate how our theory can be used for efficient evaluation of Sobolev-Fourier Coefficients

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation