Solving PDEs in C++

Yair Shapira is engaged in research in the Computer Science Department, Technion – Israel Institute of Technology, Haifa, Israel. His main research interests are multigrid, preconditioning and numerical methods. He is author of the books Matrix-Based Multigrid: Theory and Applications, 2nd Edition (2008) and Mathematical Objects in C++: Computational Tools in a Unified Object-Oriented Approach (2009).

List of figures; List of tables; Preface; Part I. Elementary Background in Programming: 1. Concise introduction to C; 2. Concise introduction to C++; 3. Data structures used in the present algorithms; Part II. Object-Oriented Programming: 4. From Wittgenstein–Lacan's theory to the object-oriented implementation of graphs and matrices; 5. FFT and other algorithms in numerics and cryptography; 6. Object-oriented analysis of nonlinear ordinary differential equations; Part III. Partial Differential Equations and their Discretization: 7. The convection-diffusion equation; 8. Some stability analysis; 9. About nonlinear conservation laws; 10. Application in image processing; Part IV. The Finite Element Discretization Method: 11. About the weak formulation; 12. Some background in linear finite elements; 13. Unstructured finite-element meshes; 14. Adaptive mesh refinement; 15. Towards high-order finite elements; Part V. The Numerical Solution of Large Sparse Linear Systems of Algebraic Equations: 16. Sparse matrices and their object-oriented implementation; 17. Iterative methods for the numerical solution of large sparse linear systems of algebraic equations; 18. Towards parallelism; Part VI. Applications in Two Spatial Dimensions: 19. Diffusion equations; 20. The linear elasticity equations; 21. The Stokes equations; 22. Application in electromagnetic waves; 23. Multigrid for nonlinear equations and for the fusion problem in image processing; Part VII. Applications in Three Spatial Dimensions: 24. Polynomials in three independent variables; 25. The Helmholtz equation: error estimate; 26. Adaptive finite elements in three spatial dimensions; 27. Application in nonlinear optics: the nonlinear Helmholtz equation in three spatial dimensions; 28. High-order finite elements in three spatial dimensions; 29. Application in the nonlinear Maxwell equations; 30. Towards inverse problems; 31. Application in the Navier–Stokes equations; Appendix. Solutions to selected exercises; Bibliography; Index.