Lec17

# Lec17 - Numerical Solution of Elliptic PDEs Finite...

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Numerical Solution of Elliptic PDEs Finite Differences, Spectral Methods, Direct and Relaxation Schemes

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Lecture 17 2 Elliptic PDEs z The problem of solving elliptic PDEs can be viewed as a higher dimensional version of the BV problems that we studied in Lectures 14-15 z Two of the most frequently encountered elliptic PDEs are: “Laplace’s Equation” “Poisson’s Equation” z These are solved inside some domain D in 2 or 3 dimensions and subject to BCs applied on a surface D that bounds D z The function f in Poisson’s equation is prescribed ; u is the unknown D D
Lecture 17 3 Forms of the Laplacian Operator z Cartesian Coordinates, u = u(x,y,z): z Polar Coordinates (2D), u = u(r, θ ) z Spherical Coordinates (3D), u = u(r, θ , φ ) 2D: 3D:

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Lecture 17 4 Typical Boundary Conditions z The classification of the BCs is analogous to 1D BV problems: Dirichlet BCs Specify the field u on the boundary D Neumann BCs Specify the outward normal derivative u/ n on the boundary D D D D D u specified u/ n specified n
Lecture 17 5 Physical Examples of Laplace and Poisson Equations z Laplace’s equation arises in many different physical contexts: Steady state temperature distribution Steady state mass distribution z Poisson’s equation is also frequently encountered: Electrostatics Steady pressure-driven viscous flow

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Lecture 17 6 Elliptic Equations: The Problem z We shall focus on the Poisson equation, since Laplace’s equation is a special case of f =0 z Discretization and approximation of derivatives by finite differences or spectral methods leads to a linear system of the form: z [L M ] is an M x M matrix, where M = N d is the number of grid points, d = # dimensions, and N = # grid points per dimension z In 3D, M can be very large! z [L M ] is generally a sparse matrix, but is not banded z As before, the boundary conditions are incorporated within this system of equations by conditions that relate field grid points on the boundary “curse of dimensionality”
Lecture 17 7 Methods for Solving Elliptic Equations z Direct Methods Direct attack on [L]{u}={f} – O(M

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Lec17 - Numerical Solution of Elliptic PDEs Finite...

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