Accurate Numerical Solution of Convection-Diffusion Problems. Vol. 1

This book consists of two volumes and is concerned with the results obtained during carrying out the project 'Accurate Numerical Solution of Convection-Diffusion Problems' of the Volkswagen Foundation.

The first volume is devoted to the results concerning the projective-difference methods of approximation of the convective-diffusion equations with convection dominated and the projective-difference methods of increasing accuracy for the second-order self-ajoint elliptic equations.

For specialists in computational mathematics.

Introduction
1	One-dimensional convection-diffusion problem
	1.1	The differential problem and its properties
		1.1.1	Boundary layer
		1.1.2	The asymptotic expansion of the solution
		1.1.3	The estimates of the remainder term
		1.1.4	The weak formulation. The Petrov-Galerkin method
	1.2	The finite element method with a linear quadrature rule
		1.2.1	Construction of the quadrature rule
		1.2.2	Properties of the discrete problem
		1.2.3	Convergence result
	1.3	The finite element method with nonlinear quadrature rule
		1.3.1	Construction of the quadrature rule
		1.3.2	Properties of the discrete problem
		1.3.3	Convergence theorem
2	Two-dimensional convection-diffusion problem
	2.1	General remarks
		2.1.1	Qualitative behaviour of the solution
		2.1.2	The weak formulation
	2.2	The scheme with the fitted quadrature rule for a problem without parabolic boundary layers
		2.2.1	The differential problem
		2.2.2	Construction of the quadrature rule
		2.2.3	Properties of the discrete problem. The convergence result
	2.3	Construction of the method for the problem with regular and parabolic boundary layers
		2.3.1	Properties of the differential problem
		2.3.2	Construction of the fitted quadrature rule
		2.3.3	The properties of the discrete problem
3	Numerical solution of the discrete problem
	3.1	Numerical experiments in the one-dimensional case
	3.2	Test example in the two-dimensional case
	3.3	The grids
	3.4	Methods for solving the discrete problem
	3.5	Discussion of the numerical results
References

Introduction
1	Some recommendations on choice of the function of steps
2	Fragmentation of the boundary of multiply connected domain
3	Triangulation of a domain
4	Conclusion
5	Appendix 1
6	Appendix 2
7	Appendix 3
8	Appendix 4
9	Appendix 5
10	Appendix 6
11	Appendix 7
12	Appendix 8
References

Introduction
1	An algorithm of determination of partial derivatives
2	Construction of a sequence of embedded grids
3	Program realization of the algorithm

Introduction
1	The difference problem statement
2	The difference approximation of convective item on an arbitrary trianqular stencil
3	Construction of inverse-monotone second-order finite-difference scheme
4	The algorithm for the orientation strengthening of the difference grid
5	The numerical experiment
References

Introduction
1	Boundary-value problem and its nonuniform difference approximation
2	Stability and solvability of the grid problem
3	Convergence of the nonuniform difference scheme
4	Numerical examples
References

Introduction
1	Boundary-value problem
2	Construction of the difference grid and classification of its nodes
3	Interpolation formula
4	Construction of difference approximation
5	Stability, solvability and convergence of the grid problem
6	Numerical examples
References

Introduction
1	Formulation of the differential problems
2	Tested methods
	2.1	Five-point scheme and Richardson extrapolation
	2.2	Nonhomogeneous Bykova-Shaidurov scheme
	2.3	Khoromskij combination
	2.4	Nine-point box scheme
3	Two ways to compare the computational cost
References