Finite element method
Illustrative problems P1 and P2

The following two problems demonstrate the finite element method.

P1 is a one-dimensional problem P1 : { u ″ (x) = f (x) in (0, 1), u (0) = u (1) = 0, {\displaystyle {\text{ P1 }}:{\begin{cases}u''(x)=f(x){\text{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}} where f {\displaystyle f} is given, u {\displaystyle u} is an unknown function of x {\displaystyle x}, and u ″ {\displaystyle u''} is the second derivative of u {\displaystyle u} with respect to x {\displaystyle x}.

P2 is a two-dimensional problem (Dirichlet problem) P2 : { u x x (x, y) + u y y (x, y) = f (x, y) in Ω, u = 0 on ∂ Ω, {\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\text{ in }}\Omega,\\u=0&{\text{ on }}\partial \Omega,\end{cases}}}

where Ω {\displaystyle \Omega } is a connected open region in the (x, y) {\displaystyle (x,y)} plane whose boundary ∂ Ω {\displaystyle \partial \Omega } is nice (e.g., a smooth manifold or a polygon), and u x x {\displaystyle u_{xx}} and u y y {\displaystyle u_{yy}} denote the second derivatives with respect to x {\displaystyle x} and y {\displaystyle y}, respectively.

The problem P1 can be solved directly by computing antiderivatives. However, this method of solving the boundary value problem (BVP) works only when there is one spatial dimension. It does not generalize to higher-dimensional problems or problems like u + V ″ = f {\displaystyle u+V''=f}. For this reason, we will develop the finite element method for P1 and outline its generalization to P2.

Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM.

After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem is then implemented on a computer.
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