The Finite Element Method in 2D
In this chapter we develop finite element methods for numerical solution of partial differential equations in two dimensions. The approach taken is the same as before, that is, we first rewrite the equation in variational form, and then seek an approximat
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The Finite Element Method in 2D
Abstract In this chapter we develop finite element methods for numerical solution of partial differential equations in two dimensions. The approach taken is the same as before, that is, we first rewrite the equation in variational form, and then seek an approximate solution in the space of continuous piecewise linear functions. Although the numerical methods presented are general, we focus on linear second order elliptic equations with the Poisson equation as our main model problem. We prove basic error estimates, discuss the implementation of the involved algorithms, and study some examples of application.
4.1 Green’s Formula At the outset let us recall a few mathematical preliminaries that will be of frequent use. Let ˝ be a domain in R2 , with boundary @˝ and exterior unit normal n. We recall the following form of the divergence theorem. Z ˝
@f dx D @xi
Z f ni ds;
i D 1; 2
(4.1)
@˝
where ni is component i of n. Setting f D fg we get the partial integration formula Z ˝
@f gdx D @xi
Z f ˝
@g dx C @xi
Z fgni ds;
i D 1; 2
(4.2)
@˝
Applying (4.2) with f D wi , the components of a vector field w on ˝, and g D v, and taking the sum over i D 1; 2 we obtain
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__4, © Springer-Verlag Berlin Heidelberg 2013
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4 The Finite Element Method in 2D
Z
Z
Z
.r w/v dx D ˝
w rv dx C
.w n/vds
˝
(4.3)
@˝
Finally, choosing w D ru in (4.3) we obtain the so-called Green’s formula Z
Z
Z
uv dx D
ru rv dx
˝
˝
n ruvds
(4.4)
@˝
We remark that Green’s formula also holds in higher space dimensions.
4.2 The Finite Element Method for Poisson’s Equation 4.2.1 Poisson’s Equation Let us consider Poisson’s equation: find u such that u D f; in ˝ u D 0;
(4.5a)
on @˝
(4.5b)
where D @2 =@x12 C @2 =@x22 is the Laplace operator, and f is a given function in, say, L2 .˝/.
4.2.2 Variational Formulation To derive a variational formulation of Poisson’s equation (4.5) we multiply f D u by a function v, which is assumed to vanish on the boundary, and integrate using Green’s formula. Z
Z f vdx D ˝
uv dx ˝
Z
(4.6) Z
ru rv dx
D Z
˝
D
n ruvds
(4.7)
@˝
ru rv dx
(4.8)
˝
The last line follows due to the assumption v D 0 on @˝. Introducing the spaces V D fv W kvkL2 .˝/ C krvkL2 .˝/ < 1g V0 D fv 2 V W vj@˝ D 0g
(4.9) (4.10)
4.2 The Finite Element Method for Poisson’s Equation
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we have the following variational formulation of (4.5): find u 2 V0 such that Z
Z ru rv dx D
f v dx;
˝
8v 2 V0
(4.11)
˝
R R With this choice of test and trial space V0 the integrals ˝ ru rv dx and ˝ f v dxR make sense. To see this, note that due to the Cauchy-Schwarz inequality, we have ˝ f v dx kf kL2 .˝/ kvkL2 .˝/ , which is less than R infinity by the assumptions on v and f . A similar line of reasoning applies to ˝ ru rv dx. In this context we would like to a point out a
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