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Numerical Methods for Elliptic and Parabolic PDEs

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1. Introduction and preliminaries

Maximum principles. Fundamental solution. Cauchy problem in a strip and in the half-space with several boundary conditions isolation, transmission, mixed. Duhamel principle. Numerical Methods: Discretization of the heat equation with finite differences. Implicit and explicit time marching schemes, the theta-method, stability analysis.

Mathematical Methods: Harmonic functions. Mean value properties.

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Newtonian potentials. Numerical Methods: Discretization with finite differences of a one-dimensional elliptic problem. Imposition of the Dirichlet and Neumann boundary conditions. Algebraic formulation and matrix properties. Diffusion-convection and diffusion-reaction problems. Mathematical Methods: String equations.

Well-posed problems and separation of variables. Numerical Methods: Discretization of the wave equation with finite difference explicit and implicit schemes.

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Leapfrog and Newmark schemes. Stability properties.

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Second Part — Functional Analysis, variational formulations and discretizations via finite element method. Mathematical Methods: Lebesgue integral. Projection theorem and Riesz representation theorem. Schwartz distributions. Sobolev spaces.

Numerical Methods: Bilinear form, abstract variational problems and Lax-Milgram lemma. Variational formulation of elliptic problems and applications to transport-reaction-diffusion equations. Introduction to the Galerkin method for a one-dimensional elliptic problem.

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  4. Consistency, stability and convergence. Cea' Lemma. The finite elements method. Linear and quadratic finite elements. Definition of Lagrangian basis functions, of composite interpolation and error estimates.

    Extension to the 2D case. Approximation of the diffusion-convection-reaction problem: comparison with the finite difference case and stability analysis. Stabilization with the upwind strategy and the mass lumping technique. Numerical Methods: Approximation with the Galerkin method, the semi-discrete problem. Actually, we realized very quickly that it would be more complicated than what it seemed at first glance, for several reasons: 1. The first version of Numerical Methods for Nonlinear Variational Problems was, in fact, part of a set of monographs on numerical mat- matics published, in a short span of time, by the Tata Institute of Fun- mental Research in its well-known series Lectures on Mathematics and Physics; as might be expected, the first version systematically used the material of the above monographs, this being particularly true for Lectures on the Finite Element Method by P.

    This second version had to be more self-contained. This necessity led to some minor additions in Chapters I-IV of the original version, and to the introduction of a chapter namely, Chapter Y of this book on relaxation methods, since these methods play an important role in various parts of this book. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.