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| Preface to the Dover Edition | |
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| Preface | |
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| Introduction | |
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| Background | |
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| Difference methods - Finite element methods | |
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| Scope of the book | |
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| Introduction to FEM for elliptic problems | |
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| Variational formulation of a one-dimensional model problem | |
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| FEM for the model problem with piecewise linear functions | |
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| An error estimate for FEM for the model problem | |
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| FEM for the Poisson equation | |
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| The Hilbert spaces L2(�), H1 (�) and H10(�) | |
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| A geometric interpretation of FEM | |
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| A Neumann problem. Natural and essential boundary conditions | |
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| Remarks on programming | |
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| Remarks on finite element software | |
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| Abstract formulation of the finite element method for elliptic problems | |
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| Introduction. The continuous problem | |
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| Discretization. An error estimate | |
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| The energy norm | |
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| Some examples | |
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| Some finite element spaces | |
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| Introduction. Regularity requirements | |
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| Some examples of finite elements | |
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| Summary | |
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| Approximation theory for FEM. Error estimates for elliptic problems | |
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| Introduction | |
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| Interpolation with piecewise linear functions in two dimensions | |
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| Interpolation with polynomials of higher degree | |
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| Error estimates for FEM for elliptic problems | |
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| On the regularity of the exact solution | |
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| Adaptive methods | |
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| An error estimate in the L2(�)-nom | |
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| Some applications to elliptic problems | |
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| The elasticity problem | |
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| Stokes problem | |
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| A plate problem | |
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| Direct methods for solving linear systems of equations | |
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| Introduction | |
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| Gaussian elimination. Cholesky's method | |
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| Operation counts. Band matrices | |
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| Fill-in | |
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| The frontal method | |
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| Nested dissection | |
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| Minimization algorithms. Iterative methods | |
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| Introduction | |
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| The gradient method | |
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| The conjugate gradient method | |
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| Preconditioning | |
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| Multigrid methods | |
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| Work estimates for direct and iterative methods | |
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| The condition number of the stiffness matrix | |
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| FEM for parabolic problems | |
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| Introduction | |
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| A one-dimensional model problem | |
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| Semi-discretization in space | |
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| Discretization in space and time | |
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| Background | |
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| The backward Euler and Crank-Nicolson methods | |
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| The discontinuous Galerkin method | |
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| Error estimates for fully discrete approximations and automatic time and space step control | |
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| Hyperbolic problems | |
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| Introduction | |
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| A convection-diffusion problem | |
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| General remarks on numerical methods for hyperbolic equations | |
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| Outline and preliminaries | |
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| Standard Galerkin | |
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| Classical artificial diffusion | |
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| The streamline diffusion method | |
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| The streamline diffusion method with �=0 | |
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| The streamline diffusion method with �>0 | |
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| The discontinuous Galerkin method | |
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| The streamline diffusion method for time-dependent convection-diffusion problems | |
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| Friedrichs' systems | |
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| The continuous problem | |
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| The standard Galerkin method | |
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| The streamline diffusion method | |
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| The discontinuous Galerkin method | |
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| Second order hyperbolic problems | |
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| Boundary element methods | |
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| Introduction | |
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| Some integral equations | |
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| An integral equation for an exterior Dirichlet problem using a single layer potential | |
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| An exterior Dirichlet problem with double layer potential | |
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| An exterior Neumann problem with single layer potential | |
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| Alternative integral equation formulations | |
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| Finite element methods | |
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| FEM for a Fredholm equation of the first kind | |
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| FEM for a Fredholm equation of the second kind | |
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| Mixed finite element methods | |
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| Introduction | |
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| Some examples | |
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| Curved elements and numerical integration | |
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| Curved elements | |
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| Numerical integration (quadrature) | |
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| Some non-linear problems | |
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| Introduction | |
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| Convex minimization problems | |
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| The continuous problem | |
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| Discretizations | |
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| Numerical methods for convex minimization problems | |
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| A non-linear parabolic problem | |
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| The incompressible Euler equations | |
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| The continuous problem | |
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| The streamline diffusion method in (�, �)-formulation | |
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| The discontinuous Galerkin method in (�, �)-formulation | |
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| The streamline diffusion method in (u, p)-formulation | |
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| The incompressible Navier-Stokes equations | |
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| Compressible flow: Burgers' equation | |
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| References | |
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| Index | |