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Unformatted text preview: 10/25/2011 1 Elliptic Problem 2 Finite Element Method Finite Elements as an Alternative to Finite Difference Method • The finiteelement method has become one of the major strategies for solving partial differential equations. • As an illustration, we develop a version of the finiteelement method for Poisson’s equation 2 = 2 ? 2 + 2 ? 2 = ?, ? over R in a two−dimensional plane. 10/25/2011 2 Minimum Energy Formulation • Solving Poisson’s equation is equivalent to (which will be proved in the future) minimizing the expression = 1 2 ? ¡ + ? ¡ + ¢£¢?. • This means that if the function minimizes the expression above, then obeys Poisson’s equation ¡ = ¡ £ ¡ + ¡ ? ¡ = £, ? ¤ over R in a two−dimensional plane. Divide and Conquer: Divide • Suppose the region is subdivided into triangles ¥ ¤ (such that ¦ =∪ ¥ and ¥ ∩ ¥ ′ = ∅...
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 Fall '11
 Xli
 Finite Element Method, Partial differential equation, finite elements, two−dimensional plane

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