Contents -- 0 Models of incompressible fluid flow -- 1 The Poisson equation -- 1.1 Reference problems -- 1.2 Weak formulation -- 1.3 The Galerkin finite element method -- 1.3.1 Triangular finite elements (R[sup(2)]) -- 1.3.2 Quadrilateral elements (R[sup(2)]) -- 1.3.3 Tetrahedral elements (R[sup(3)]) -- 1.3.4 Brick elements (R[sup(3)]) -- 1.4 Implementation aspects -- 1.4.1 Triangular element matrices -- 1.4.2 Quadrilateral element matrices -- 1.4.3 Assembly of the Galerkin system -- 1.5 Theory of errors -- 1.5.1 A priori error bounds -- 1.5.2 A posteriori error bounds -- 1.6 Matrix properties -- Problems -- Computational exercises -- 2 Solution of discrete Poisson problems -- 2.1 The conjugate gradient method -- 2.1.1 Convergence analysis -- 2.1.2 Stopping criteria -- 2.2 Preconditioning -- 2.3 Singular systems are not a problem -- 2.4 The Lanczos and minimum residual methods -- 2.5 Multigrid -- 2.5.1 Two-grid convergence theory -- 2.5.2 Extending two-grid to multigrid -- Problems -- Computational exercises -- 3 The convection-diffusion equation -- 3.1 Reference problems -- 3.2 Weak formulation and the convection term -- 3.3 Approximation by finite elements -- 3.3.1 The Galerkin finite element method -- 3.3.2 The streamline diffusion method -- 3.4 Theory of errors -- 3.4.1 A priori error bounds -- 3.4.2 A posteriori error bounds -- 3.5 Matrix properties -- 3.5.1 Computational molecules and Fourier analysis -- 3.5.2 Analysis of difference equations -- Discussion and bibliographical notes -- Problems -- Computational exercises -- 4 Solution of discrete convection-diffusion problems -- 4.1 Krylov subspace methods -- 4.1.1 GMRES -- 4.1.2 Biorthogonalization methods -- 4.2 Preconditioning methods and splitting operators -- 4.2.1 Splitting operators for convection-diffusion systems -- 4.2.2 Matrix analysis of convergence.

4.2.3 Asymptotic analysis of convergence -- 4.2.4 Practical considerations -- 4.3 Multigrid -- 4.3.1 Practical issues -- 4.3.2 Tools of analysis: smoothing and approximation properties -- 4.3.3 Smoothing -- 4.3.4 Analysis -- Discussion and bibliographical notes -- Problems -- Computational exercises -- 5 The Stokes equations -- 5.1 Reference problems -- 5.2 Weak formulation -- 5.3 Approximation using mixed finite elements -- 5.3.1 Stable rectangular elements (Q[sub(2)]-Q[sub(1)], Q[sub(2)]-P[sub(-1)], Q[sub(2)]-P[sub(0)]) -- 5.3.2 Stabilized rectangular elements (Q[sub(1)]-P[sub(0)], Q[sub(1)]-Q[sub(1)]) -- 5.3.3 Triangular elements -- 5.3.4 Brick and tetrahedral elements -- 5.4 Theory of errors -- 5.4.1 A priori error bounds -- 5.4.2 A posteriori error bounds -- 5.5 Matrix properties -- 5.5.1 Stable mixed approximation -- 5.5.2 Stabilized mixed approximation -- Discussion and bibliographical notes -- Problems -- Computational exercises -- 6 Solution of discrete Stokes problems -- 6.1 The preconditioned MINRES method -- 6.2 Preconditioning -- 6.2.1 General strategies for preconditioning -- 6.2.2 Eigenvalue bounds -- 6.2.3 Equivalent norms for MINRES -- 6.2.4 MINRES convergence analysis -- Discussion and bibliographical notes -- Problems -- Computational exercises -- 7 The Navier-Stokes equations -- 7.1 Reference problems -- 7.2 Weak formulation and linearization -- 7.2.1 Stability theory and bifurcation analysis -- 7.2.2 Nonlinear iteration -- 7.3 Mixed finite element approximation -- 7.4 Theory of errors -- 7.4.1 A priori error bounds -- 7.4.2 A posteriori error bounds -- Discussion and bibliographical notes -- Problems -- Computational exercises -- 8 Solution of discrete Navier-Stokes problems -- 8.1 General strategies for preconditioning -- 8.2 Approximations to the Schur complement operator.

8.2.1 The pressure convection-diffusion preconditioner -- 8.2.2 The least-squares commutator preconditioner -- 8.3 Performance and analysis -- 8.3.1 Ideal versions of the preconditioners -- 8.3.2 Use of iterative methods for subproblems -- 8.3.3 Convergence analysis -- 8.3.4 Enclosed flow: singular systems are not a problem -- 8.3.5 Relation to SIMPLE iteration -- 8.4 Nonlinear iteration -- Discussion and bibliographical notes -- Problems -- Computational exercises -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W.