Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug04 667

Problem 5a State Newton's method for the approximate solution of where is a real-valued function of the real variable Solution 5a Problem 5b State and prove a convergence result for the method. Solution 5b Problem 5c What is the typical order of convergence? Are there situations in which the order of convergence is higher? Explain your answers to these questions. Solution 5c The typical order of local convergence is quadratic. Consider the Newton's method as a fixed point iteration i.e.: Then Expanding around gives an expression for the error Note that if , then we have better than quadratic convergence. Problem 6 Consider the boundary value problem Problem 6a Derive a variational formulation for (1). Solution 6a Find such that for all Problem 6b What do we mean by Finite Element Approximation to Solution 6b Let be a partition of . Choose a an appropriate discrete subspace and basis functions . Then The coefficients can be found by solving the following system of equations: For Problem 6c State and prove an estimate for Solution 6c Cea's Lemma: In particular choose to be the linear interpolant of . Then, Alternative Solution 6c Let be a discrete mesh of with step size . Consider the following integral . For some , as is just a linear interpolation on this interval. Hence . Similarly, we can bound the norm of the error in the derivatives with . With such intervals we have Problem 6d Prove the formula Solution 6d Problem 7 Consider the initial value problem where satisfies the Lipschitz condition for all . A numerical method called the midpoint rule for solving this problem is defined by where is a time step and for . Here is given and is presumed to be computed by some other method. Problem 7a Suppose the problem is posed on a finite interval where . Show directly,i.e., without citing any major results, that the midpoint rule is stable. That is show that if and satisfy then there exists a constant independent of such that for Solution 7a Subtracting both equations, letting , and applying the Lipschitz property yields, Therefore, Problem 7b Suppose instead we are interested in the long term behavior of the midpoint rule applied to a particular example . That is, let be fixed and let so that the rule is applied over a long time interval. Show that in this case the midpoint rule does not produce an accurate approximation to the solution to the initial value problem. Solution 7b Substituting into the midpoint rule we have, or The solution of this equation is given by where or the roots of the quadratic The quadratic formula yields If is a small negative number, than one of the roots will be greater than 1. Hence, as instead of converging to zero since .