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Numerical Methods for Partial Differential Equations (SMA 5212)

Graduate · Historical course

Spring 2003

Topics

  • Overview
  • Finite Differences: Elliptic Problems
  • Finite Differences: Parabolic Problems
  • Finite Differences: Eigenvalue, 2D Problems
  • Solution Methods: Iterative Methods
  • Solution Methods: Multigrid Methods
  • Finite Differences: Hyperbolic Problems
  • Finite Volumes: Linear Problems
  • Finite Volumes: Conservation Laws
  • Finite Volumes: Nonlinear Problems
  • Finite Elements: Variational Formulation
  • Finite Elements: Poisson 1D – I
  • Finite Elements: Poisson 1D – II
  • Finite Elements: Poisson 2D – I
  • Finite Elements: Poisson 2D – II
  • Finite Elements: General Elliptic Problems – Overview
  • Finite Elements: Parabolic Problems, Eigenvalue Problems
  • Integral Equations: Derivation
  • Integral Equations: Collocation and Galerkin Methods
  • Integral Equations: Convergence Theory – 2nd Kind
  • Integral Equations: Quadrature and Cubature
  • Integral Equations: Nystrom Methods
  • Integral Equations: Convergence Theory – 1st Kind
  • Integral Equations: Fast Solvers

Readings

  • Trefethen, L. N., and D. Bau, III. Numerical Linear Algebra. Philadelphia, PA: SIAM, 1997. ISBN: 9780898713619.
    Listed reading
  • Numerical Methods for Conservation Laws , R. Levecque, Lectures in Mathematics, ETH Zurich, Birkhauser.
    Listed reading
  • Strang, G., and G. J. Fix. Analysis of the Finite Element Method . Upper Saddle River, NJ: Prentice-Hall, 1973. ISBN: 9780130329462.
    Listed reading
  • Quarteroni, A., and A. Valli. Numerical Approximation of Partial Differential Equations . Berlin; New York, NY: Springer-Verlag, 1997. ISBN: 9783540571117 (Berlin: acid-free paper) and ISBN: 9780387571119 (New York: acid-free paper).
    Listed reading
  • Atkinson, K. E. The Numerical Solution of Integral Equations of the Second Kind . Cambridge, UK: Cambridge University Press, 1997. ISBN: 9780521583916 (hardcopy).
    Listed reading
  • Briggs, W. L., et al. A Multigrid Tutorial . 2nd ed. Philadelphia, PA: SIAM, 2000. ISBN: 9780898714623.
    Listed reading
  • Tveito, A., and R. Winther. Introduction to Partial Differential Equations: A Computational Approach . New York, NY: Springer, 1998. ISBN: 9780387983271. (Texts in Applied Mathematics 29, are on reserve at MIT’s Barker Library and at NUS’s Central Library and SMA Library.)
    Listed reading
Sources