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Unformatted text preview: Lecture 39 Finite Difference Method for Elliptic PDEs Examples of Elliptic PDEs Elliptic PDEs are equations with second derivatives in space and no time derivative. The most important examples are Laplaces equation u = u xx + u yy + u zz = 0 and the Poisson equation u = f ( x, y, z ) . These equations are used in a large variety of physical situations such as: steady state heat problems, steady state chemical distributions, electrostatic potentials, elastic deformation and steady state fluid flows. For the sake of clarity we will only consider the two dimensional problem. A good model problem in this dimension is the elastic deflection of a membrane. Suppose that a membrane such as a sheet of rubber is stretched across a rectangular frame. If some of the edges of the frame are bent, or if forces are applied to the sheet then it will deflect by an amount u ( x, y ) at each point ( x, y ). This u will satify the boundary value problem: u xx + u yy = f ( x, y ) for ( x, y ) in R, u ( x, y ) = g ( x, y ) for ( x, y ) on R, (39.1) where R is the rectangle, R is the edge of the rectangle, f ( x, y ) is the force density (pressure)...
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