Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Jan04 667

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Problem 4a[edit]

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

Solution 4a[edit]

From Taylor expansion of and around , we have



Let be a uniform partition of with step size


Then for we have



For



For


Problem 4b[edit]

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.

Solution 4b[edit]


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.


Therefore


Then the finite difference method and the finite element method yield the same matrix.

Problem 4c[edit]

Show that the matrix in is non singluar.

Solution 4c[edit]

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 .

Problem 5[edit]

Consider the following dissipative initial value problem,



where is smooth and satisfies


Problem 5a[edit]

Write the Backward Euler Method for (2). This gives rise to an algebraic equation. Explain how you would solve this equation.

Solution 5a[edit]

Using Taylor Expansion we have



Thus we have Backwards Euler Method:



Let

Problem 5b[edit]

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.

Solution 5b[edit]

Subtracting and , we have


Problem 6[edit]


Solution 6[edit]