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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2014 • Question 6

Question

Math 6610 • Analysis of Numerical Methods I Fall 2014 • Question 6

Consider the variational principle: \[ I(u)=\int _0^1 \left (u'(x)\right )^2+k^2\left (u(x)\right )^2 dx=\min \qquad \text {subject to}\qquad u(0)=u(1)=1. \] Suppose you want to use the Ritz Method to approximate the solution of this problem by a polynomial of degree \(n\). Derive a linear system for the coefficients of that polynomial.

Raw LaTeX

Consider the variational principle:
\[
  I(u)=\int_0^1 \left(u'(x)\right)^2+k^2\left(u(x)\right)^2 dx=\min
  \qquad \text{subject to}\qquad u(0)=u(1)=1.
\]
Suppose you want to use the Ritz Method to approximate the solution of this
problem by a polynomial of degree \(n\). Derive a linear system for the
coefficients of that polynomial.

Topic Tags

Discrete Maximum Principles Iterative Methods for Linear Systems Linear Programming and Optimization Polynomial Interpolation and Error Formulas