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| Models of incompressible fluid flow | |
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| The Poisson equation | |
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| Reference problems | |
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| Weak formulation | |
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| The Galerkin finite element method | |
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| Triangular finite elements (R[superscript 2]) | |
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| Quadrilateral elements (R[superscript 2]) | |
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| Tetrahedral elements (R[superscript 3]) | |
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| Brick elements (R[superscript 3]) | |
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| Implementation aspects | |
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| Triangular element matrices | |
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| Quadrilateral element matrices | |
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| Assembly of the Galerkin system | |
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| Theory of errors | |
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| A priori error bounds | |
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| A posteriori error bounds | |
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| Matrix properties | |
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| Problems | |
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| Computational exercises | |
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| Solution of discrete Poisson problems | |
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| The conjugate gradient method | |
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| Convergence analysis | |
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| Stopping criteria | |
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| Preconditioning | |
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| Singular systems are not a problem | |
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| The Lanczos and minimum residual methods | |
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| Multigrid | |
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| Two-grid convergence theory | |
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| Extending two-grid to multigrid | |
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| Problems | |
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| Computational exercises | |
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| The convection-diffusion equation | |
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| Reference problems | |
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| Weak formulation and the convection term | |
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| Approximation by finite elements | |
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| The Galerkin finite element method | |
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| The streamline diffusion method | |
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| Theory of errors | |
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| A priori error bounds | |
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| A posteriori error bounds | |
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| Matrix properties | |
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| Computational molecules and Fourier analysis | |
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| Analysis of difference equations | |
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| Discussion and bibliographical notes | |
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| Problems | |
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| Computational exercises | |
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| Solution of discrete convection-diffusion problems | |
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| Krylov subspace methods | |
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| GMRES | |
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| Biorthogonalization methods | |
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| Preconditioning methods and splitting operators | |
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| Splitting operators for convection-diffusion systems | |
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| Matrix analysis of convergence | |
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| Asymptotic analysis of convergence | |
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| Practical considerations | |
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| Multigrid | |
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| Practical issues | |
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| Tools of analysis: smoothing and approximation properties | |
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| Smoothing | |
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| Analysis | |
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| Discussion and bibliographical notes | |
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| Problems | |
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| Computational exercises | |
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| The Stokes equations | |
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| Reference problems | |
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| Weak formulation | |
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| Approximation using mixed finite elements | |
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| Stable rectangular elements (Q[subscript 2]-Q[subscript 1], Q[subscript 2]-P[subscript -1], Q[subscript 2]-P[subscript 0]) | |
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| Stabilized rectangular elements (Q[subscript 1]-P[subscript 0], Q[subscript 1]-Q[subscript 1]) | |
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| Triangular elements | |
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| Brick and tetrahedral elements | |
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| Theory of errors | |
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| A priori error bounds | |
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| A posteriori error bounds | |
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| Matrix properties | |
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| Stable mixed approximation | |
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| Stabilized mixed approximation | |
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| Discussion and bibliographical notes | |
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| Problems | |
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| Computational exercises | |
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| Solution of discrete Stokes problems | |
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| The preconditioned MINRES method | |
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| Preconditioning | |
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| General strategies for preconditioning | |
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| Eigenvalue bounds | |
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| Equivalent norms for MINRES | |
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| MINRES convergence analysis | |
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| Discussion and bibliographical notes | |
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| Problems | |
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| Computational exercises | |
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| The Navier-Stokes equations | |
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| Reference problems | |
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| Weak formulation and linearization | |
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| Stability theory and bifurcation analysis | |
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| Nonlinear iteration | |
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| Mixed finite element approximation | |
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| Theory of errors | |
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| A priori error bounds | |
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| A posteriori error bounds | |
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| Discussion and bibliographical notes | |
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| Problems | |
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| Computational exercises | |
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| Solution of discrete Navier-Stokes problems | |
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| General strategies for preconditioning | |
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| Approximations to the Schur complement operator | |
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| The pressure convection-diffusion preconditioner | |
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| The least-squares commutator preconditioner | |
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| Performance and analysis | |
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| Ideal versions of the preconditioners | |
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| Use of iterative methods for subproblems | |
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| Convergence analysis | |
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| Enclosed flow: singular systems are not a problem | |
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| Relation to SIMPLE iteration | |
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| Nonlinear iteration | |
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| Discussion and bibliographical notes | |
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| Problems | |
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| Computational exercises | |
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| Bibliography | |