Introduction to Numerical Methods for Partial Differential Equations
MATH 410

Winter 2014
Assignment 2 Solution, MA 410/510, Winter 2014
1. (a) Use the Maximum Principle to show that if u satises
u + 4u = sin2 (u) in (0, 1) ,
u(0) = u(1) = 0 ,
then u 0 on [0, 1].
(b) Show that u = x(1 x)(1 2x) satises
au + bu = x < 0 in (0, 1) ,
u(0) = u(1) =
Introduction to Numerical Methods for Partial Differential Equations
MATH 410

Winter 2014
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Introduction to Numerical Methods for Partial Differential Equations
MATH 410

Winter 2014
Assignment 1, MA 410/510, Winter 2014
1. It can be shown that the Laplacian, =
coordinates as
u =
1
r2 r
r2
u
r
+
2
x2
2
2
+ y2 + z2 , can be expressed in spherical
1
2 sin
r
sin
u
+
1
2u
,
r2 sin2 2
where x = r sin cos , y = r sin sin , z = r cos . Us
Introduction to Numerical Methods for Partial Differential Equations
MATH 410

Winter 2014
Examples of Convergence Rates
Uniformly rened triangulations cfw_Th
N is number of triangles, so h2 N 1
Vh is piecewise linear space on Th , with appropriate B.C.
E0 = u uh 0 , E1 = u uh 1
Problem with smooth solution: u = x(1 x)y (1 y )
u=0
u = 2
Introduction to Numerical Methods for Partial Differential Equations
MATH 410

Winter 2014
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