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

Abstract

The free boundary problem for a fully penetrating well of radius r and filled with water to a depth h_w, 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 (ku_x)_x + (ku_y)_y =0 in Ω, u(R,y) = H for 0 < y < H, u_n(x,0) = 0 for r < x < R, u(r,y) = h_w for 0 < y < h_w, u(r,y) = y for h_w < y < ϕ (r), u_n(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.