Math 6610 • Analysis of Numerical Methods I

{Finite-Element Methods}
Consider the problem
\[
  -u''+u=f,\qquad \text{on }[0,1]
\]
with \(u(0)=0\) and \(u(1)=1\).

Derive a variational formulations of this problem (be sure to define the space on
which your variational problem is to be solved). Formulate a Galerkin
finite-element method for the variational problem using piecewise linear functions.
Prove that the linear system which arises in the finite-element method has a
solution. In what sense is the solution optimal?

\fact{1} A real symmetric \(n\times n\) matrix \(A\) can be diagonalized by an
orthogonal similarity transformation, and \(A\)'s eigenvalues are real.

\fact{2} The \((N-1)\times(N-1)\) matrix \(M\) defined by
\[
  m_{ij}=
  \begin{cases}
    2, & i=j,\\
    -1, & \lvert i-j\rvert=1,\\
    0, & \text{otherwise},
  \end{cases}
\]
has eigenvalues
\[
  \mu_l=4\sin^2\left(\frac{\pi l}{2N}\right),\qquad l=1,2,\ldots,N-1.
\]

\fact{3} The \((N+1)\times(N+1)\) matrix:
\[
  \begin{bmatrix}
    -1 & 1 & 0 & \cdots & 0 & 0\\
    1 & -2 & 1 & \ddots & 0 & 0\\
    0 & 1 & -2 & \ddots & 0 & 0\\
    \vdots & \ddots & \ddots & \ddots & 1 & 0\\
    0 & 0 & 0 & 1 & -2 & 1\\
    0 & 0 & 0 & 0 & 1 & -1
  \end{bmatrix}
\]
has eigenvalues
\[
  \mu_l=-4\sin^2\left(\frac{\pi l}{2(N+1)}\right),\qquad l=0,1,\ldots,N.
\]

\fact{4} For a real \(n\times n\) matrix \(A\), the Rayleigh quotient of a vector
\(x\in\R^n\) is the scalar
\[
  r(x)=\frac{x^TAx}{x^Tx}.
\]
The gradient of \(r(x)\) is
\[
  \nabla r(x)=\frac{2}{x^Tx}(Ax-r(x)x).
\]
If \(x\) is an eigenvector of \(A\) then \(r(x)\) is the corresponding eigenvalue and
\(\nabla r(x)=0\).