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In the last few years there has been a growing interest in the development of numerical techniques appropriate for the approximation of differential model problems presenting multiscale solutions. This is the case, for instance, with functions displaying a smooth behavior, except in certain regions where sudden and sharp variations are localized. Typical examples are internal or boundary layers. When the number of degrees of freedom in the discretization process is not sufficient to ensure a fine resolution of the layers, some stabilization procedures are needed to avoid unpleasant oscillatory effects, without adding too much artificial viscosity to the scheme. In the field of finite elements, the streamline diffusion method, the Galerkin least-squares method, the bub­ ble function approach, and other recent similar techniques provide excellent treatments of transport equations of elliptic type with small diffusive terms, referred to in fluid dynamics as advection-diffusion (or convection-diffusion) equations. Goals This book is an attempt to guide the reader in the construction of a computa­ tional code based on the spectral collocation method, using algebraic polyno­ mials. The main topic is the approximation of elliptic type boundary-value par­ tial differential equations in 2-D, with special attention to transport-diffusion equations, where the second-order diffusive terms are strongly dominated by the first-order advective terms. Applications will be considered especially in the case where nonlinear systems of partial differential equations can be re­ duced to a sequence of transport-diffusion equations.

Inhaltsverzeichnis

Frontmatter

1. The Poisson Equation in the Square

Abstract
We begin by examining the numerical approximation, using the Legendre collocation method, of a simple boundary-value problem defined in a square. This analysis is the starting point for several generalizations to be developed in the following chapters.
Daniele Funaro

2. Steady Transport-Diffusion Equations

Abstract
To more general elliptic boundary-value problems we apply the numerical technique introduced in the previous chapter. Namely, we are interested in differential operators in which the second-order terms are dominated by the first-order terms, giving rise to the boundary layer phenomenon. To better deal with these equations, a modified version of the Legendre collocation grid will be considered.
Daniele Funaro

3. Other Kinds of Boundary Conditions

Abstract
We study the polynomial approximation of boundary-value problems of elliptic type subjected to Neumann or mixed-type boundary conditions. As we did in the previous chapters, we develop a preconditioned iterative method for a fast and accurate resolution of the corresponding linear systems.
Daniele Funaro

4. The Spectral Element Method

Abstract
We analyze the discretization of elliptic boundary-value problems defined in domains with a complicated shape, via a domain decomposition approach. The approximated solution is a patchwork of different algebraic polynomials defined in the subdomains and is determined as the result of a preconditioned iterative procedure. At any iteration, in any subdomain, the corresponding polynomial satisfies a collocation problem.
Daniele Funaro

5. Time Discretization

Abstract
We turn our attention to the approximation of partial differential equations where the solutions also depend on time. We discuss some well-known boundary-value problems and consider their discretization both in space and time.
Daniele Funaro

6. Extensions

Abstract
We present some other results and open questions related to what has been developed in the previous chapters. The study and improvement of these ideas could be the subject of future investigations.
Daniele Funaro

Backmatter

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