Introduction - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 25

4.6 Example. Suppose we want to find the solution of the potential equation in
the unit disk satisfying the (Dirichlet) boundary condition

∞

1
k(k −

u( cos ϕ, sin ϕ) =

1 ) cos kϕ.

k = 2

Since it is absolutely and uniformly convergent, the series represents a continuous
function. By (1.2), the solution of the boundary-value problem in polar coordinates
is

∞

r k
k(k −

u(x, y) =

1 ) cos kϕ.

( 4 . 7 )

k = 2

Now on the x -axis, the second derivative

∞

1

x k − 2

u xx (x, 0 ) = =

1 − x

k = 2

is unbounded in a neighborhood of the boundary point ( 1 , 0 ) , and thus Theorem 4.5
is not directly applicable.
A complete convergence theory can be found, e.g., in Hackbusch [1986]. It
uses the stability of the differential operator in the sense of the L 2 -norm, while
here the maximum norm was used (but see Problem 4.8). Since the main topic
of this topic is the finite element method, we restrict ourselves here to a simple
generalization. Using an approximation-theoretical argument, we can extend the
convergence theorem at least to a disk with arbitrary continuous boundary values.
By the Weierstrass approximation theorem, every periodic continuous func-
tion can be approximated arbitrarily well by a trigonometric polynomial. Thus, for
given ε> 0, there exists a trigonometric polynomial

m

v( cos ϕ, sin ϕ) = a 0 + (a k cos kϕ + b k sin kϕ)

k = 1

ε
4

with | v − u | < on ∂. Let

m

r k (a k cos kϕ + b k sin kϕ)

v(x,y) = a 0 +

k =

1

and let V be the numerical solution obtained by the finite difference method. By
the maximum principle and the discrete maximum principle, it follows that

ε
4

ε
4

| u − v | < in , | U − V | < in h . ( 4 . 8 )