Consider the boundary value problem
where , and . Formulate a difference method for the approximate solution of on a uniform mesh of size . Explain how is approximated by a difference quotient
From Taylor expansion of and around , we have
Let be a uniform partition of with step size
Then for we have
Suppose and in . Formulate a finite element method for the approximate solution of in this special case, again on a uniform mesh. Using the standard "hat functions" basis for the finite element space, write out the finite element equations explicitly. Show that if an appropriate quadrature formula is used on the right-hand side of the finiite element equations, they (the finite element equations) are the same as the finite difference equations.
Since we are integrating hat functions on the right hand side, an appropriate quadrature formula would be to take half of the midpoint rule. The regular midpoint rule would give double the actual integral value of a hat function.
Then the finite difference method and the finite element method yield the same matrix.
Show that the matrix in is non singluar.
Since the matrix is diagonally dominant, it is non-singular.
To show that the matrix has a non-zero determinant, 2n elementary row operation can be used to show that
has the same determinant as
which is .
Consider the following dissipative initial value problem,
where is smooth and satisfies
Write the Backward Euler Method for (2). This gives rise to an algebraic equation. Explain how you would solve this equation.
Using Taylor Expansion we have
Thus we have Backwards Euler Method:
Derive an error estimate of the form
where . Do this directly, not as an application of a standard theorem. (Note that there is no exponential on the right hand side.
Subtracting and , we have