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41 using a galerkin formulation with a piecewise

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Unformatted text preview: 1 8v 2 H0 : Solve this problem on a sequence of ner and ner grids using piecewise linear, quadratic, and cubic nite element bases. Select a basic grid with either two or four elements in it and obtain ner grids by uniform re nement of each element into four elements. Present plots of the energy error as functions of the number of degrees of freedom (DOF), the mesh spacing h, and the CPU time for the three polynomial bases. De ne h as the square root of the area of an average element. 5.5. Element Matrices and Their Assembly 35 You may combine the convergence histories for the three polynomial solutions on one graph. Thus, you'll have three graphs, error vs. h, error vs. DOF, and error vs. CPU time, each having results for the three polynomial solutions. Estimate the convergence rates of the solutions. Comment on the results. Are they converging at the theoretical rates? Are there any unexpected anomalies? If so, try to explain them. You may include plots of solutions and/or meshes to help answer these questions. 6. Consider the Dirichlet problem for Laplace's equation u = uxx + uyy = 0 (x y) 2 u(x y) = (x y) (x y) 2 @ where is the L-shaped region with lines connecting the Cartesian vertices (0,0), (1,0), (1,1), (-1,1), (-1,-1), (0,-1), (0,0). Select (x y) so that the exact solution expressed in polar coordinates is u(r ) = r2=3 sin 23 : with x = r cos y = r sin : This solution has a singularity in its rst derivative at r = 0. The singularity is typical of those associated with the solution of elliptic problems at re-entrant corners such as the one found at the origin. Because of symmetries, the problem need only be solved on half of the L-shaped domain, i.e., the trapezoidal region ~ with lines connecting the Cartesian vertices (0,0), (1,0), (1,1), (-1,1), (0,0). 1 The Galerkin form of this problem consists of determining u 2 HE ZZ vxux + vy uy ]dxdy = 0 1 8v 2 H0 : ~ 1 Functions u 2 HE satisfy the essential boundary conditions u(x y) = 0 y=0 0<x<1 x=1 0 y<1 y=1 ;1 < x...
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