Unformatted text preview: cessary to ensure the existence
of integrals and solution accuracy. The use of piecewise polynomial functions simplify
the evaluation of integrals involved in the L2 inner product and strain energy (1.2.2b,
1.2.9b) and help automate the solution process. Choosing bases with small support leads
to a sparse, well-conditioned linear algebraic system (1.2.12c)) for the solution.
Let us illustrate the nite element method by solving the two-point boundary value
problem (1.2.1) with constant coe cients, i.e.,
;pu 00 + qu = f (x) 0<x<1 u(0) = u(1) = 0 (1.3.1) where p > 0 and q 0. As described in Section 1.2, we construct a variational form of
(1.2.1) using Galerkin's method (1.2.11). For this constant-coe cient problem, we seek
to determine u 2 H01 satisfying A(v u) = (v f )
(v u) = A(v u) = Z 1 1
8v 2 H0 Z1
0 (v pu + vqu)dx:
0 0 vudx
(1.3.2c) 1.3. A Simple Finite Element Problem 9 With u and v belonging to H01, we are sure that the integrals (1.3.2b,c) exist and that
the trivial boundary conditions are satis ed.
We will subsequently show that func...
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