Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug07 667

Problem 4 Consider the problem of solving a nonlinear system of ODE by an implicit method. The -th step consists of solving for the unknown a non-linear algebraic system of the form where is known, and is the stepsize. Let Problem 4a Write as a fixed point iteration and find conditions in and that local convergence for this iteration ononon Solution 4a Fixed point iteration Equation is conveniently in fixed point iteration form. Notice that the right hand side is only a function of since are fixed when solving for the fixed point where Also note that is the fixed point iteration index. Conditions for local convergence The fixed point iteration will converge when the norm of the Jacobian of is less than 1 i.e. Since , we equivalently have the condition Problem 4b Write the Newton iteration for and give conditions on and that guarantee local convergence for this iteration. State precise additional assumptions on that guarantee quadratic convergence Solution 4b Newton iteration The Newton iteration solves and the iteration is given by Let Conditions for local convergence If exists, i.e. is invertible or equivalently non-singular, then local convergence is guaranteed. Note that Conditions for quadratic convergence If is Lipschitz, then we have quadratic convergence and is twice continuously differentiable in a neighborhood of the root Problem 5 This problem is about choosing between a specific single-step and a specific multi-step methods for solving ODE: Problem 5a Write the trapezoid method, define its local truncation error and estimate it. Solution 5a Trapezoid method (Implicit, Adams-Moulton) Define Local Truncation Error The local truncation error is given as follows: Find Local Truncation Error Using Taylor Expansion Note that . The uniform step size is . Hence, Therefore, the given equation may be written as Expand Left Hand Side Expanding about , we get Expand Right Hand side Also expanding about gives Calculate local truncation error Since the order 3 terms of do not agree (), the error is of order . Problem 5b Show that the truncation error for the following multistep method is of the same order as in (a): Solution 5b We need to show that Again, note that So this method is also consistency order 2. Problem 5c What could be said about the global convergence rate for these two methods? Justify your conclusions for both methods. Solution 5c The trapezoid is stable because its satisfies the root condition. (The root of the characteristic equation is 1 and has a simple root) The second method is not stable because the characteristic equation has a double root of 1. Both the trapezoid method and second method are consistent with order Note that convergence occurs if and only the method is both stable and consistent. Therefore, the trapezoid method converges in general but the second method does not. mkmkmkmlmklml Problem 6 Consider the boundary value problem where is constant. Let be a uniform meshsize . Let be the corresponding finite element space, and let be the corresponding finite element solution of (2). Note that is a projection operator, the Ritz projector, onto the finite dimensional space with respect to the element scalar product induced by problem (2). Problem 6a Let be the -seminorm, namely for all . Find the constant in terms of the parameter such that Hint: recall the Poincare inequality for all where denotes the -norm Solution 6a Weak Form Integrating by parts gives, for all Specifically, Discretized Form (Finite Element Formulation) Similarly, the finite element formulation is find such that for all Specifically, Equate Both Sides and Apply Inequalities Hence we have, Problem 6b If is the Lagrange interpolant of , then prove . Deduce Solution 6b Prove equality We have for all Specifically, for all The discrete form of the energy scalar product is for all Subtracting equation (2) from equation (1), we have Let . Note that by hypothesis . Then, By ellipticity, which implies i.e. Deduce inequality Hence we have Arguing as we did in part (a), we have Problem 6c Use (b) to derive the error estimate and bound the right hand side by suitable power of . Make explicit the required regularity of Solution 6c Show inequality Bound Right Hand Side For , Newton's polynomial interpolation error gives for some Therefore the error on the entire interval is given by which implies needs to be twice differentiable.