How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace – and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal...

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace—and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Ga...

Sparse tensor product spaces provide an efficient tool to discretize higher dimensional operator equations. The direct Galerkin method in such ansatz spaces may employ hierarchical bases, interpolets, wavelets or multilevel frames. Besides, an alternative approach is provided by the so-called combination technique. It properly combines the Galerkin solutions of the underlying problem on certain...

This paper presents an application of the nonlinear optimal control techniques to the design of launch vehicle autopilots. The optimal control is given by the solution to the Hamilton-Jacobi-Bellman (HJB) equation, which in this case cannot be solved explicity. A method based upon Successive Galerkin Approximation (SGA), is used to obtain an approximate optimal solution. Simulation results invo...

— Families of C piecewise polynomial spaces of degree r ^ 3 on triangles and quadrilatéral in two dimensions are constructed, and approximation properties of the families are studied. Examples of the use of the families in Galerkin methodsfor 2nd and 4th order elliptic boundary value problems on arbitrarily shaped domains are given. The approximation properties on the boundary are such that the...

a truly meshless local petrov-galerkin (mlpg) method is developed for solving 3d elasto-static problems. using the general mlpg concept, this method is derived through the local weak forms of the equilibrium equations, by using test functions, namely, the heaviside function. the moving least squares (mls) are chosen to construct the shape functions, for the mlpg method. the penalty approximatio...

The present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method ( GM ), Galerkin method with weak form ( / GM WF ), Petrov-Galerkin method ( PGM ), weighted residual method (WRY ), and least squares method or process ( LSM or LSP ) to construct finite element approximations in time. A correspond...

Space-time variational formulations of infinite-dimensional Fokker–Planck (FP) and Ornstein–Uhlenbeck (OU) equations for functions on a separable Hilbert space H are developed. The well-posedness of these equations in the Hilbert space L2(H,μ) of functions on H, which are square-integrable with respect to a Gaussian measure μ on H, is proved. Specifically, for the infinite-dimensional FP equati...

This paper analyzes a method for solving the thirdand fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov–Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials P (α,β) n with α, β ∈ (−1, ∞) and n is the polynomial degree. By choosing appropriate base functions, the resulting system ...

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n 1 2 −r) in infinity norm, and the iterated Legendre Galerkin ...