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u(r ) = r2=3 sin 23
These boundary conditions may be expressed in Cartesian coordinates by using
r 2 = x2 + y 2
tan = x : 36 Mesh Generation and Assembly
The solution of the Galerkin problem will also satisfy the natural boundary condition un = u = 0 along the diagonal y = ;x.
Solve this problem using available nite element software. To begin, create a
three-element initial mesh by placing lines between the vertices (0,0) and (1,1)
and between (0,0) and (0,1). Generate ner meshes by uniform re nement and use
piecewise-polynomial bases of degrees one through three.
As in Problem 5, present plots of the energy error as functions of the number of
degrees of freedom, the mesh spacing h, and the CPU time for the three polynomial bases. You may combine the convergence histories for the three polynomial
solutions on one graph. De ne h as the square root of the area of an average element. Estimate the convergence rates of the solutions. Is accuracy higher with a
high-order method on a coarse mesh or with a low-order method on a ne mesh?
If adaptivity is available, use a piecewise-linear basis to calculate a solution using
adaptive h-re nement. Plot the energy error of this solution with those of the
uniform-mesh solutions. Is the adaptive solution more e cient? E ciency may
be de ned as less CPU time or fewer degrees of freedom for the same accuracy.
Contrast the uniform and adaptive meshes. 5.6 Assembly of Vector Systems
Vector systems of partial di erential equations may be treated in the same manner as the
scalar problems described in the previous section. As an example, consider the vector
version of the model problem (5.5.1): determine u 2 HE satisfying A(v u) = (v f )
(v f ) = A(v u) = ZZ ZZ 1
8v 2 H0 v f dxdy (5.6.1b) T v Pu + v Pu + v Qu]dxdy:
x x T
y y (5.6.1a) T (5.6.1c) The functions u(x y), v(x y), and f (x y) are m-vectors and P and Q are m m matrices.
Smooth solutions of (5.6.1) satisfy
;(Pux )x ; (Puy )y + Qu = f (x y) 2 (5.6.2a) 5.5. Assembly of Vector Systems u= 37 (x y...
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