The Finite Element Method in 1D
In this chapter we shall introduce the finite element method as a general tool for the numerical solution of two-point boundary value problems. In doing so, the basic idea is to first rewrite the boundary value problem as a variational equation, and then
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The Finite Element Method in 1D
Abstract In this chapter we shall introduce the finite element method as a general tool for the numerical solution of two-point boundary value problems. In doing so, the basic idea is to first rewrite the boundary value problem as a variational equation, and then seek a solution approximation to this equation from the space of continuous piecewise linears. We prove basic error estimates and show how to use these to formulate adaptive algorithms that can be used to automatically improve the accuracy of the computed solution. The derivation and areas of application of the studied boundary value problems are also discussed.
2.1 The Finite Element Method for a Model Problem 2.1.1 A Two-point Boundary Value Problem Let us consider the following two-point boundary value problem: find u such that u00 D f;
x 2 I D Œ0; L
u.0/ D u.L/ D 0
(2.1a) (2.1b)
where f is a given function. Sometimes this problem is easy to solve analytically. For example, if f D 1, then we readily find u D x.L x/=2 by integrating f twice and using the boundary conditions u.0/ D u.L/ D 0. However, for a general f it may be difficult or even impossible to find u with analytical techniques. Thus, we see that even a very simple differential equation like this one may be difficult to solve analytically. We take this as a good motivation for introducing the finite element method, which is a general numerical technique for solving differential equations.
M.G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Texts in Computational Science and Engineering 10, DOI 10.1007/978-3-642-33287-6__2, © Springer-Verlag Berlin Heidelberg 2013
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2 The Finite Element Method in 1D
2.1.2 Variational Formulation The derivation of a finite element method always starts by rewriting the differential equation under consideration as a variational equation. This so-called variational formulation is in our case obtained by multiplying f D u00 by a test function v, which is assumed to vanish at the end-points of the interval I , and integrating by parts. Z L Z L f v dx D u00 v dx (2.2) 0
0
Z
L
D Z
u0 v0 dx u0 .L/v.L/ C u0 .0/v.0/
(2.3)
u0 v0 dx
(2.4)
0 L
D 0
The last line follows from the assumption v.0/ D v.L/ D 0. For this calculation to make sense we must assert that the test function v is not too badly behaved so that the involved integrals do indeed exist. More specific, we require that both v and v0 be square integrable on I . Of course, v must also vanish at x D 0 and x D L. Now, the largest collection of functions with these properties is given by the function space V0 D fv W kvkL2 .I / < 1; kv0 kL2 .I / < 1; v.0/ D v.L/ D 0g
(2.5)
Obviously, this space contains many functions, and any of them can be used as test function v. In fact, there are infinitely many functions in V0 , and we therefore say that V0 has infinite dimension. Not just v, but also u, is a member of V0 . To see this, note that u is twice differentiable, which implies that u0 is smooth, and satisf
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