University of Oulu
INFOTECH OULU

Infotech Oulu Graduate School

Numerical solution of boundary value problems

Lecturer: Professor Alexander Lapin, Kazan State University, Russia

First lecture: Thursday, October 8, 2009, at 10-12, room: M201.

Credits: 10 ects (56h + exercises).

Dates & times: on Mondays 12-14, on Thursdays, 10-12, on Fridays 12-14

Room: M201

Contents

Part I. Getting started: Approximation of two-point boundary value problems (8-10 h.).

  1. A model problem: Dirichlet boundary-value problem (BVP) for one-dimensional diffusion equation. Finite difference method (FDM). Construction, stability and convergence analysis.
  2. Finite element method for model problem. Variational formulation of the problem. Construction and analysis of FEM.
  3. Approximation of 1D problem with variable coefficients. Using quadrature formulas in FEM. Relationship between finite differences and finite elements.
  4. Approximation of the 1D convection-diffusion equation. Construction of FDM and FEM. Numerical viscosity, stabilized methods.
  5. Exercises.

Part II. Approximation of linear equations in Hilbert spaces (6-8 h.).

  1. Linear equations in Hilbert spaces. Equations with bounded and positive definite operators. Case of self-adjoint operator and minimisation problem.
  2. Weak solutions of the elliptic BVP. Quick introduction to Sobolev spaces theory. Weak formulations of the second order equations with Dirichlet, Neumann and Robin boundary conditions.
  3. Ritz and Galerkin methods. Perturbed Galerkin method. Formulations. Convergence. Error bounds. Cea lemma. Strang lemma.
  4. Exercises.

Part III. FDM and FEM for the 2-nd order elliptic equations (14-16 h.).

  1. FDMfor Poisson equation. Construction, stability and convergence analysis. Error bounds.
  2. Construction of FEM for the elliptic equations with variable coefficients. Lagrange triangle and quadrangle finite elements. Reference finite elements. Construction of the stiffness matrices and the load vectors.
  3. Convergence and error bounds for FEM. Interpolation error estimates in Sobolev spaces. Error bounds by Cea lemma and interpolation error estimates.
  4. Construction of FEM with quadrature formulas for the equations with variable coefficients. Multidimensional quadrature formulas. Construction of FEM. Existence and uniqueness of a solution. Interpretation of the classical FDM as FEM with quadrature formulas.
  5. Convergence and error bounds for FEM with quadrature formulas. Error bounds by Strang lemma and error estimates for quadrature formulas.
  6. Exercises.

Part IV. Solution methods for linear systems (14-16 h.).

  1. Overview of direct methods. Gauss elimination and LU factorisation. Inverse of a tridiagonal matrix. Cholesky factorisation for symmetric and positive definite matrices. Cuthill-McKee algorithm for reordering the system variables.
  2. Stationary iterative methods. Richardson method. Jacobi, Gauss-Seidel and relaxation iterative methods.
  3. Non-stationary iterative methods. Tchebychev method. Conjugate gradient method. GMRES for nonsymmetric systems.
  4. Preconditioning matrices. Generalities. Jacobi and SSOR preconditioners. Incomplete factorisation preconditioners.
  5. Exercises.

Part V. FDM and FEM for the 2-nd order parabolic equations (8-10 h.).

  1. FDM for the heat equation. Explicit and implicit approximation in time. Stability analysis by the spectral and energy methods. Convergence and error estimates.
  2. FDM and FEM for the 2-nd order parabolic equations with variable coefficients. Stability analysis by energy methods. Convergence and error estimates.
  3. Exercises.

Learning outcomes: On successful completion of this course, the student will be able to

  • construct correct finite difference and finite element methods for linear partial differential equations of second order;
  • implement efficient numerical algorithms for solving large systems of linear equations with sparse matrices;
  • construct and implement the most typical numerical algorithms for solving linear time-dependent problems.

References

[1] O. Ladyzhenskaya. The boundary value problems of mathematical
physics, N.Y.: Springer Verlag, 1985 .

[2] R. Adams. Sobolev spaces, N.Y.: Academic Press, 1975.

[3] A. Quarteroni, R. Sacco, F. Saleri. Numerical mathematics (Texts in applied mathematics; 37), N.Y.: Springer Verlag, 2000.

[4] Ph. Ciarlet, J.-L. Lions. Handbook of mumerical analysis: finite element
methods, Amsterdam, North-Holland, 1991.

[5] A. Samarskii, E. Nikolaev. Numerical methods for grid equations. V.I: Direct methods. V. II: Iterative methods, Bazel, Birkhauser Verlag, 1989.

[6] W. Hackbush. Iterative solution of large sparse systems of equations, N.Y.: Springer Verlag, 1994.

[7] O. Axelsson. Iterative solution methods, N.Y.: Cambridge University Press, 1996.

[8] Y. Saad. Iterative methods for sparse linear systems. Second edition, Philadelphia, PA: SIAM, 2003.

More information: Erkki Laitinen

Infotech Oulu Graduate School Courses