Approximation Of Evolution Equations Research Articles

Asymptotic Equivalence of Evolution Equations Governed by Cocoercive Operators and Their Forward Discretizations

The purpose of this work is to study discrete approximations of evolution equations governed by cocoercive operators by means of Euler iterations, both in a finite and in an infinite time horizon. On the one hand, we give precise estimations for the distance between iterates of independently generated Euler sequences and use them to obtain bounds for the distance between the state, given by the continuous-time trajectory, and the discrete approximation obtained by the Euler iterations. On the other hand, we establish the asymptotic equivalence between the continuous- and discrete-time systems, under a sharp hypothesis on the step sizes, which can be removed for operators deriving from a potential. As a consequence, we are able to construct a family of smooth functions for which the trajectories/sequences generated by basic first-order methods converge weakly but not strongly, extending the counterexample of Baillon. Finally, we include a few guidelines to address the problem in smooth Banach spaces.

Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns

The IMG algorithm (Inertial Manifold-Multigrid algorithm) which uses the first-order incremental unknowns was introduced in [20]. The IMG algorithm is aimed at numerically implementing inertial manifolds (see e.g. [19]) when finite difference discretizations are used. For that purpose it is necessary to decompose the unknown function into its long wavelength and its short wavelength components; (first-order) Incremental Unknowns (IU) were proposed in [20] as a means to realize this decomposition. Our aim in the present article is to propose and study other forms of incremental unknowns, in particular the Wavelet-like Incremental Unknowns (WIU), so-called because of their oscillatory nature. In this report, we first extend the general convergence results in [20] by proving them under slightly weaker conditions. We then present three sets of incremental unknowns (i.e. the first-order as in [20], the second-order and wavelet-like incremental unknowns). We show that these incremental unknown can be used to construct convergent IMG algorithms. Special stress is put on the wavelet-like incremental unknowns since this set of unknowns has theL 2 orthogonality property between different levels of unknowns and this should make them particularly appropriate for the approximation of evolution equations by inertial algorithms.

Nonlinear Galerkin methods: The finite elements case

With the increase in the computing power and the advent of supercomputers, the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem. In this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements. The algorithm that we propose, called the nonlinear Galerkin method, stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete (finite elements) space. Essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases. Beside a detailed description of the algorithm, the article includes some technical results on finite elements spaces, and a full study of the stability and convergence of the method.

On uniform difference schemes of a high order of approximation for evolution equations with a small parameter

We consider initial value problems for evolution equations in Banach space E with a small parameter E multiplying the time-derivative term. For such problems we construct difference schemes of a high order of approximation that are uniform with respect to the parameter in (0,1).

Approximation of evolution equations with polynomial reproducing nonlinearities

We study evolution equations of typeut=Au+Nu with polynomial nonlinearityN, in a Banach spaceB which lies in a Hilbert space. Under the restriction that the nonlinear operatorN is “finitely reproducing relative to the orthonormal sequenceei generated byAu=λu, the explicitly known Faedo-Galerkin approximations of the evolution equation can be estimated. The “reproducing” property is shown in the special case of a diffusion equation with Neumann boundary conditions and a nonlinearity of third degree. We study the numerical behavior of the approximations.