Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug06 667

Problem 4 Suppose that is smooth and that the boundary value problem has unique solution. Problem 4a For , let . Write down a system of equations to obtain an approximation for the solution at by replacing the second derivatives by a symmetric difference quotient. Solution 4a The symmetric difference quotient is given by Hence we have the following system equations that incorporates the initial conditions . Problem 4b Write the system of equations in the form . Define domain and range of and explain the meaning of the variable . Solution 4b Domain: Range: is a vector containing approximations for the solution at Problem 4c Formulate Newton's method for the solution of the system in (b) with . Give explicit expressions for all objects involved (as far as this is reasonable). Determine a sufficient condition that ensures that the iterates in the Newton scheme are defined. Without doing any further calculations, can you decide whether the sequence converges. Why or why not? Solution 4c Newton's Method where denotes the Jacobian of a matrix . Specifically, Sufficient Condition If exists, then iterates are defined. Convergence of sequence We cannot decide if the sequence converges since Newton's method only guarantees local convergence. In general, for local convergence of Newton's method we need: differentriable invertible Lipschitz close to solution Problem 5 Consider the boundary value problem with boundary conditions and . Here is a given positive number. Problem 5a Describe a Galerkin method to solve this problem using piecewise linear functions with respect to a uniform mesh. Weak Formulation Find such that for all which after integrating by parts and plugging in initial conditions we have Let be the nodes of a uniform partition of where and . Let be the standard "hat" functions defined as follows: For Also since Then forms a basis for the discrete space Problem 5b Derive the matrix equations for this Galerkin method. Write out explicitly that equation of the linear system which involves Discrete Weak Formulation Find such that for all Since forms a basis, we have Also for In matrix form Problem 6 Consider the linear multistep method for the solution of the initial value problem Problem 6a Show that the truncation error is of order 2. Problem 6b State the condition for consistency of a linear multistep method and verify it for the scheme in this problem. Solution 6b Conditions: (i) (ii) Problem 6c Does the scheme satisfy the root condition and or the strong root condition? The scheme satisfies the root condition but not the strong root condition since the roots are given by which implies and