Abstract
AbstractWe consider the transformation of quasilinear partial differential equations to a coupled system of linear equations, but with nonlinear transmission conditions on the interfaces. After deriving a variational formulation, we will discuss Mortar finite element discretization strategies and present a numerical example. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Highlights
This model problem can be seen as a simplification of the stationary Richards equation without gravity, see [1]
Such a simplification can not be achieved if kr depends in addition explicitly on x ∈ Ω
The idea is to introduce Lagrange multipliers acting on interfaces which ensure global continuity of the solution
Summary
As a model problem we consider the following quasilinear boundary value problem in a bounded domain Ω ⊂ Rd (d = 2, 3) with Lipschitz boundary Γ = ∂Ω = ΓD ∪ ΓN and ΓD ∩ ΓN = ∅ to find p such that. −∆u = f in Ω, u = 0 on ΓD, ∇u · n = gN on ΓN Such a simplification can not be achieved if kr depends in addition explicitly on x ∈ Ω. If this dependency is piecewise within non–overlapping subdomains Ωi, we can introduce local Kirchhoff transformations ui(x) := κi(pi(x)) for x ∈ Ωi. We can reformulate problem (1) and obtain the following nonlinear transmission problem, in the case of two subdomains, see Figure 1, to find ui, i = 1, 2, such that. We will derive a variational formulation for the nonlinear transmission problem (2)
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