Lyashko I. I.
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The numerical-analytic solution of boundary-value problems for the equation div(η grad ϕ) = F with piecewise-constant η
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The theoretical principles of the method of majorant regions are outlined in this paper, and its application to the solution of concrete problems of the plane theory of filtration in homogeneous and heterogeneous media is discussed. In § 1 the problem of a two-grooved floodbed with deep-set apron at various bottom marks of the upstream and downstream water in a permeable layer of infinite depth is solved for the case of a homogeneous medium, and the problem of the determination of the discharge through an earthwork weir with mattress type drainage on a permeable base of finite depth is also solved. Strict upper and lower estimates are found for the filtration characteristics, sufficiently close to each other to secure the accuracy necessary in practice. In § 2 the author considers filtration without backwater from a channel of arbitrary cross section in a porous zonally heterogeneous soil (stratified). Making use of known properties of p-analytical functions [4, 51, estimates of the discharge are found by the majorant region method. Numerical examples confirming the efficacy of the method of majorant regions for this class of problems are discussed in all cases