A general framework is presented to discuss the approximate solutions of an evolution equation in a Banach space, with a linear part generating a semigroup and a sufficiently smooth nonlinear part. A theorem is presented, allowing to infer from an approximate solution the existence of an exact solution. According to this theorem, the interval of existence of the exact solution and the distance ...

Let V be a Hilbert space, a(·, ·) : V × V → lR a bounded, V-elliptic bilinear form and : V → lR a bounded linear functional. We want to approximate the variational equation: Find u ∈ V such that (3.1) a(u, v) = (v) , v ∈ V. We recall that the Lax-Milgram Lemma ensures the existence and uniqueness of a solution of (3.1). Now, given a finite-dimensional subspace V h ⊂ V , dim V h = n h , the Ritz...

The tracking control of non-minimum phase systems is nowadays an open and challenging field, because a general theory is still not available. This article proposes an indirect control strategy in which a key role is played by the inverse problem that arises and their approximate solutions. These are obtained with the Galerkin method, a standard functional analysis tool. A detailed study of the ...

We present an explicit form for the construction of approximate inertial manifolds (AIMs) for a class of nonlinear dissipative partial differential equations by using a perturbation technique. We investigate two numerical examples of the reaction diffusion equation with polynomial nonlinearity and non-polynomial nonlinearity to show comparison of accuracy for our perturbation method with other ...

We study the order of convergence Galerkin variational integrators for ordinary differential equations. approximate a (Lagrangian) problem by restricting space curves to set polynomials degree at most $s$ and approximating action integral using quadrature rule. show that, if rule is sufficiently accurate, thus obtained $2s$.

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in [4, 5, 24]. In particular, we show that the superconvergence points for the gradient of the ap...

We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the L-orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree k + l, k + 1 and k, where k...

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order condition the discussed. Weighted Jacobi polynomials are used to approximate and adjoint state. A priori error estimates for control, state, Lagrangian multiplier derived. Numerical experiment carried ou...

In this paper, we discuss the application of Generalized Multiscale Finite Element Method (GMsFEM) to elasticity equation in heterogeneous media. Our applications are motivated by elastic wave propagation in subsurface where the subsurface properties can be highly heterogeneous and have high contrast. We present the construction of main ingredients for GMsFEM such as the snapshot space and o in...