An abstract functional framework is developed for the dual Petrov-Galerkin formulation of the initial boundary value problems with a third-order spatial derivative. This framework is then applied to study the wellposedness and decay properties of Airy equation and KdV equation in a finite interval.

We investigate an h-p version of the continuous Petrov-Galerkin method for the nonlinear Volterra functional integro-differential equations with vanishing delays. We derive h-p version a priori error estimates in the L-, Hand L-norms, which are completely explicit in the local discretization and regularity parameters. Numerical computations supporting the theoretical results are also presented.

We construct space-time Petrov–Galerkin discretizations of the heat equation on an unbounded temporal interval, either right-unbounded or left-unbounded. The discrete trial and test spaces are defined using Laguerre polynomials in time and are shown to satisfy the discrete inf-sup condition. Numerical examples are provided.

A level set method combining features of front tracking methods and fixed domain methods is presented to model microstructure evolution in the solidification of multi-component multiphase alloy systems. Phase boundaries are explicitly tracked by solving the multi-phase level set equations. Diffused interfaces are constructed by extending a small width in both directions from these explicitly tr...

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...

Non-symmetric matrices may arise in the discretization of self-adjoint problems when a Petrov–Galerkin, collocation, or finite-volume method is employed. While these methods have been widely applied, this paper it shown that use non-symmetric incompatable with conservation energy elastodynamics. First, consistency between continuous forms momentum equation and examined. It linear equivalent to ...

We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) for the two-point boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of Tk). Super-geometric convergent rate is established for a special class of solutions.

An abstract functional framework is developed for the dual Petrov-Galerkin formulation of the initial-boundary-value problems with a third-order spatial derivative. This framework is then applied to study the wellposedness and decay properties of the KdV equation in a finite interval.

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...

This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key ingredient for a successful reduced order solution to inverse problems is the consistency of the reduced order Karush-KuhnTucker conditions with respect to the full optimality conditions. In particular, a...