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In this text, we introduce the basic concepts for the numerical modeling of partial differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws, saddle-point problems and optimal control problems. Furthermore, we provide numerous physical examples which underline such equations.
-A brief survey of partial differential equations
-Elements of functional analysis
-Elliptic equations
-The Galerkin finite element method for elliptic problems
-Parabolic equations
-Generation of 1D and 2D grids
-Algorithms for the solution of linear systems
-Elements of finite element programming
-The finite volume method
-Spectral methods
-Isogeometric analysis
-Discontinuous element methods (DG and mortarmethods)
-Diffusion-transport-reaction equations
-Finite differences for hyperbolic equations
-Finite elements and spectral methods for hyperbolic equations
-Nonlinear hyperbolic problems
-Navier-Stokes equations
-Optimal control of partial differential equations
-Domain decomposition methods
-Reduced basis approximation for parametrized partial differential equations