Davide Guidetti

Approximation of Abstract Differential Equations

This review paper is devoted to the numerical analysis of abstract differential equations in Banach spaces. Most of the finite difference, finite element, and projection methods can be considered from the point of view of general approximation schemes (see, e.g., [207,210,211] for such a representation). Results obtained for general approximation schemes make the formulation of concrete numerical methods easier and give an overview of methods which are suitable for different classes of problems. The qualitative theory of differential equations in Banach spaces is presented in many brilliant papers and books. We can refer to the bibliography [218], which contains about 3000 references. Unfortunately, no books or reviews on general approximation theory appear for differential equations in abstract spaces during last 20 years. Any information on the subject can be found in the original papers only. It seems that such a review is the first step towards describing a complete picture of discretization methods for abstract differential equations in Banach spaces. In Sec. 2 we describe the general approximation scheme, different types of convergence of operators, and the relation between the convergence and the approximation of spectra. Also, such a convergence analysis can be used if one considers elliptic problems, i.e., the problems which do not depend on time. Section 3 contains a complete picture of the theory of discretization of semigroups on Banach spaces. It summarizes Trotter–Kato and Lax–Richtmyer theorems from the general and common point of view and related problems. The approximation of ill-posed problems is considered in Sec. 4, which is based on the theory of approximation of local C-semigroups. Since the backward Cauchy problem is very important in applications and admits a stochastic noise, we also consider approximation using a stochastic regularization. Such an approach was never considered in the literature before to the best of our knowledge. In Sec. 5, we present discrete coercive inequalities for abstract parabolic equations in Cτn([0, T ];En), C τn([0, T ];En), L p τn([0, T ];En), and Bτn([0, T ];C (Ωh)) spaces.

A GLOBAL IN TIME EXISTENCE AND UNIQUENESS RESULT FOR A SEMILINEAR INTEGRODIFFERENTIAL PARABOLIC INVERSE PROBLEM IN SOBOLEV SPACES

We prove a global in time abstract existence and uniqueness result for a general parabolic problem of reconstruction of a convolution kernel. The result is, in particular, applicable to the theory of heat conduction for materials with memory.

An Inverse Problem for a Phase-field Model in Sobolev Spaces

We prove an existence and uniqueness result for an inverse problem arising from a phase-field model with two memory kernels. More precisely, we identify the convolution memory kernels and the diffusion coefficient besides the temperature and the phase-field parameter. We prove our results in the framework of Sobolev spaces. Our fundamental tools are an optimal regularity result in the L p spaces and fixed point arguments.

Asymptotic expansion of solutions to an inverse problem of parabolic type with non-homogeneous boundary conditions

We consider an inverse parabolic problem of reconstruction of the source function, together with the traditional solution. In contrast with older literature, we consider non-homogeneous and time-dependent boundary conditions. We are able to prove a general result of convergence to a stationary state, and of asymptotic expansion as t → ∞.

An inverse problem for the beam equation with memory with nonhomogeneous boundary conditions

Let ? be an open bounded set in , for n = 1, 2, 3 and let T > 0. The problem we study is: determine u and h, under suitable initial and nonhomogeneous boundary conditions, satisfying the equation (for (t, x) (0, T) ? ?) and the additional restriction on u: where and G are given data. The nonlinear function F may contain spatial derivatives up to order two. We prove a global in time existence and uniqueness result for the beam equation with memory in the case where F is sublinear. In this case we only assume regularity conditions on F, but we do not assume any growth condition, we prove a local in time existence theorem and a global in time uniqueness result.

Backward Euler Scheme, Singular Hölder Norms, and Maximal Regularity for Parabolic Difference Equations

We show finite difference analogues of maximal regularity results for discretizations of abstract linear parabolic problems. The involved spaces are discrete versions of spaces of Hölder continuous functions, which can be singular in 0. The main tools are real interpolation and Da Prato–Grisvards theory of the sum of linear operators.

On a Finite Difference Scheme for an Inverse Integro-Differential Problem Using Semigroup Theory: A Functional Analytic Approach

ABSTRACT In this article, the problem of reconstructing an unknown memory kernel from an integral overdetermination in an abstract linear (of convolution type) evolution equation of parabolic type is considered. After illustrating some results of the existence and uniqueness of a solution for the differential problem, we study its approximation by Rothes method. We prove a result of stability and another concerning the order of approximation of the solution in dependence of its regularity. The main tool is a maximal regularity result for solutions to abstract parabolic finite difference schemes. Two model problems to which the results are applicable are illustrated.

Identification of a convolution kernel in a control problem for the heat equation with a boundary memory term

We consider the evolution of the temperature

Convergence to a stationary state and stability for solutions of quasilinear parabolic equations

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