Introduction - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 4

the above binomial formula, we get D(u, v) = 0. Now applying Green's integral

formula, we have

0 = D(u, v) = (u x v x + u y v y )dxdy

=− v(u xx + u yy )dxdy + v(u x dy − u y dx).

∂

The contour integral vanishes because of the boundary condition for v . The first
integral vanishes for all v ∈ C 1 () if and only if u = u xx + u yy = 0. This

proves that (1.5) characterizes the solution of the (linearized) Plateau problem.

1.4 The Wave Equation as a Prototype of a Hyperbolic Differential Equa-
tion. The motion of particles in an ideal gas is subject to the following three laws,
where as usual, we denote the velocity by v , the density by ρ , and the pressure by
p :
1. Continuity Equation.

∂ρ
∂t =− ρ 0 div v.
Because of conservation of mass, the change in the mass contained in a
(partial) volume V must be equal to the flow through its surface, i.e., it must

be equal to ∂V ρv · ndO. Applying Gauss' integral theorem, we get the above
equation. Here ρ is approximated by the fixed density ρ 0 .
2. Newton's Law.

ρ 0 ∂v

∂t =−

grad p.

The gradient in pressure induces a force field which causes the acceleration
of the particles.
3. State Equation.

p = c 2 ρ.

In ideal gases, the pressure is proportional to the density for constant temper-
ature.

Using these three laws, we conclude that

∂t 2 p = c 2 ∂ 2 ρ

∂ 2

∂t 2 =− c 2 ∂ ∂t ρ 0 div v =− c 2 div (ρ 0 ∂v

∂t )

c 2 div grad p c 2 p . = =

Other examples of the wave equation

u tt = c 2 u