The Numerical Solution of Axisymmetric Free Boundary Porous Flow Well Problems Using Variational Inequalities

  • Colin W. Cryer
  • Hans Fetter
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Serie Internationale D’Analyse Numerique book series (ISNM, volume 48)


The free boundary problem for a fully penetrating well of radius r and filled with water to a depth hw, in a layer of soil of depth H, radius R and permeability k(x,y) can be formulated as follows: Find ϕ ∈ C1[r,R] and \({\text{u}} \in {\text{C}}^2 (\Omega)\; \cap \;{\text{C(}}\overline \Omega {\text{)}}\) such that (kux)x + (kuy)y =0 in Ω, u(R,y) = H for 0 < y < H, un(x,0) = 0 for r < x < R, u(r,y) = hw for 0 < y < hw, u(r,y) = y for hw < y < ϕ (r), un(x, ϕ (x)) = 0 and u(x, ϕ (x)) = ϕ (x) for r < x < R, where Ω = { (x,y: 0 < y < ϕ (x)}. The results of Benci [Annali di Mat. 100 (1974), 191–209] are used to derive a variational inequality and to prove existence and uniqueness. The problem is approximated using piecewise linear finite elements and 0(h) convergence of the approximate solutions is proved using recent results due to Brezzi, Hager, and Raviart.


Porous Medium Variational Inequality Free Boundary Hydraulic Head Free Boundary Problem 
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Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Colin W. Cryer
    • 1
  • Hans Fetter
    • 1
  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA

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