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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.
Department of Mathematics and Computer Science
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.
- Lectures On Numerical Methods For Non-Linear Variational Problems (Scientific Computation).
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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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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.
sne77.fr/includes/joseph/rencontre-femme-outre-mer.php 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.