Accurate Numerical Solution of Convection-Diffusion Problems. Vol. 2

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 second volume is devoted, first, to multigrid for systems of linear algebraic equations obtained by finite-element method for the second-order self-adjoint elliptic equations and, second, to projective-difference methods of approximation of Navier-Stokes equations for viscous incompressible fluid.

For specialists in computational mathematics.

Introduction
1	The cascadic algorithm for 2D problems
	1.1	The weakly nonlinear elliptic equation
		1.1.1	Formulation of the differential problem
		1.1.2	Formulation of the discrete problem
		1.1.3	Formulation of the cascadic algorithm
		1.1.4	Auxiliary estimates
		1.1.5	Convergence of the cascadic algorithm
	1.2	The indefinite-sign elliptic problem
		1.2.1	Formulation of the differential problem
		1.2.2	Formulation of the discrete problem
		1.2.3	Formulation of the cascadic algorithm
		1.2.4	An auxiliary operator
		1.2.5	Convergence of the cascadic algorithm
		1.2.6	Optimization of the number of iterations
	1.3	The plane elasticity problem
		1.3.1	Formulation of the differential problem
		1.3.2	Formulation of the discrete problem
		1.3.3	Formulation of the cascadic algorithm
		1.3.4	Convergence of the cascadic algorithm and optimization of the number of iterations
2	The cascadic algorithm for the 3D Dirichlet problem
	2.1	The 3D Dirichlet problem on a polyhedron
		2.1.1	Formulation of the differential problem
		2.1.2	Formulation of the discrete problem
		2.1.3	Formulation of the cascadic algorithm
		2.1.4	Convergence of the cascadic algorithm
		2.1.5	Optimization of the number of iterations
	2.2	Asymptotic stability of the algorithm of triangulation refine ment for a 3D domain
		2.2.1	The algorithm of dividing
		2.2.2	Criteria of quality of a triangulation
		2.2.3	Estimation of quality of the triangulation
	2.3	The cascadic algorithm for a domain with a smooth curvilin ear boundary
		2.3.1	Formulation of the differential problem
		2.3.2	Formulation of the discrete problem
		2.3.3	An auxiliary result
		2.3.4	Convergence of the Galerkin solution
		2.3.5	The estimate of the eigenvalues of the matrix of the discrete problem
		2.3.6	Convergence of the cascadic algorithm
3	Numerical results
	3.1	Dependence of the convergence rate of the V-cycle upon smoothing
		3.1.1	Preliminary remarks
		3.1.2	Formulation of the multigrid algorithm
		3.1.3	Numerical tests
	3.2	Cascadic algorithm for the Poisson equation
		3.2.1	Two-step semi-iterative process
		3.2.2	Dependence of the convergence on the number of iteration steps
References

Introduction
1	The formulation of the problem and the splitting into physical processes
2	Discretization of the fractional step of pressure work
	2.1	Integration over Ω
	2.2	Integration with the help of small fictitious domains for uniformity of equations
3	Discretization of the fractional step of convection-diffusion
	3.1	Futher splitting and discretization of the equation for the first component of velocity
	3.2	Splitting and discretization of the equation for the second component of velocity
	3.3	Integration with the help of small fictitious domains for uniformity of equations
4	Numerical experiments
5	Conclusions
	5.1	Splitting method vs. solution with complete operator
	5.2	Staggerred meshes vs. united mesh
	5.3	Square vs. triangle mesh
References