Analysis of completely discrete finite element method for a free boundary diffusion problem with absorption (Q5946732)

This work is concerned with the use of truncation methods to obtain approximate solutions to a free boundary problem that is a model of the dispersion of pheromones by a species of butterfly that attacks grapes in vineyards. The problem is one of determining a function \(c\) (the concentration) that satisfies the equation NEWLINE\[NEWLINEc_t- \alpha\Delta c + \text{div} (Vc) = -f,\quad x \in \omega(t),\;t>0, \tag{1}NEWLINE\]NEWLINE where \(\omega(t)\) is a subset of a bounded domain \(\Omega\) in \(\mathbb{R}^2\) in which \(c\) takes positive values, \(V\) is the wind velocity, which satisfies the Navier-Stokes equations, \(\alpha\) is a diffusion coefficient, and \(f \geq 0\) is an absorption coefficient. The boundary conditions are \(c = 0\) on \(\partial \Omega \times (0,T)\), and \(c = \partial c / \partial n = 0\) on \(\partial \omega(t)\); the initial conditions are \(c(x,0) = c_0(x) \geq 0\), where \(c_0(x) > 0\) on \(\omega(0)\) and \(\omega (0)\) is strictly included in \(\Omega\). NEWLINENEWLINENEWLINEThe authors make use of a truncation method that extends in various ways the methods proposed in by \textit{A. E. Berger, M. Ciment}, and \textit{J. C. W. Rogers} [SIAM J. Numer. Anal. 12, 646-672 (1975; Zbl 0317.65032)]. In particular, it is assumed here that the absorption depends on space and time, and the truncation method is based on a backward Euler method in time and a conforming \(P_1\)-finite element method in space. The main goal of this work is to obtain \(L^\infty\)-error estimates for the problem at hand. NEWLINENEWLINENEWLINEThe problem is written in operator form in an obvious way as NEWLINE\[NEWLINE c_t + Ac = -fa(c) \tag{2}NEWLINE\]NEWLINE in which \(a(c)\) is the Heaviside step function. Two truncation methods, which enforce the positivity of approximations \(c_n\) in two different ways, are considered. Both are based on an Euler backward approximation together with a linearization of the term involving \(a\), and take the following forms: NEWLINENEWLINENEWLINEMethod I: NEWLINE\[NEWLINE\begin{cases} (I + \Delta t A)u^{n+1} = c^n,\\ c^{n+1}= \max (u^{n+1} - \Delta t f(t(n+1)),0).\end{cases} NEWLINE\]NEWLINE Method II: NEWLINE\[NEWLINE \begin{cases} u^n= \max (c^n - \Delta t f(t(n+1)),0), \\ (I + \Delta t A)c^{n+1}= u^n .\end{cases}NEWLINE\]NEWLINE The rest of the paper is concerned with proving the convergence of the two methods. This is achieved by first estimating the error between the exact solution and the approximations \(c_I\;(i=1,2)\) obtained by replacing \((I + \Delta t A)^{-1}\), the term appearing in the Euler backward approximation, with \(e^{-tA}\), the semigroup generated by \(A\). It is shown that, for \(f\) and its derivative that are in \(L^\infty\), this error is estimated by NEWLINE\[NEWLINE \|c(n\tau) - c_i(n\tau)\|\leq C\tau\quad \text{for} i=1,2\quad \text{and} n\tau \leq T. NEWLINE\]NEWLINE Semi-discretization in time, using a backward Euler method, leads to approximations \(c_i^{nk}\) of \(c_i\) for which the error estimate NEWLINE\[NEWLINE \|c_i^{nk} - c_i(n\tau)\|\leq C\frac{n}{k}\|c_0\|NEWLINE\]NEWLINE is obtained. Thus the error between \(c_i^{nk}\) and \(c\) may be obtained, and from this it is shown that the choice \(k = O(\Delta t^{1/3})\) minimises the error, which is then \(O(\Delta t^{1/3})\). Here \(k\) is the number of time steps after which truncation is applied. It is then shown that the same order of convergence is obtained when truncation is applied at each time step. NEWLINENEWLINENEWLINEThe final part of the paper is concerned with fully discrete approximations. Using a conforming finite element with piecewise linear approximation on triangles, various error estimates are obtained. For example, using truncation at each time step, the error for both truncation methods is \(O (\Delta t^{1/3})|\ln h|^{1/3} + h^{2/5}\), where \(h\) is the finite element mesh size, provided that the time step satisfies \(ah^2|\ln h|^{3} \leq \Delta t\).