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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 Lshaped 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 reentrant
corners such as the one found at the origin.
Because of symmetries, the problem need only be solved on half of the Lshaped
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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 Spring '14
 JosephE.Flaherty
 Geometry, Vector Space, 11:11, Boundary value problem, Boundary conditions, Dirichlet boundary condition

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