Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production
- PDF / 1,172,423 Bytes
- 23 Pages / 439.37 x 666.142 pts Page_size
- 97 Downloads / 166 Views
Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production Vo Anh Khoa1,2
· Ekeoma Rowland Ijioma3 · Nguyen Nhu Ngoc4
Received: 18 November 2019 / Revised: 8 August 2020 / Accepted: 18 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann–Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in H 1 norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove the H 1 -type contraction at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.
Communicated by Frederic Valentin. Vo Anh Khoa was funded by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044. The work of the first author was also supported by a postdoctoral fellowship of the Research Foundation-Flanders (FWO) in Belgium.
B
Nguyen Nhu Ngoc [email protected] Vo Anh Khoa [email protected] ; [email protected] Ekeoma Rowland Ijioma [email protected]
1
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
2
Faculty of Sciences, Hasselt University, Campus Diepenbeek, Agoralaan Building D, BE3590 Diepenbeek, Belgium
3
Meiji Institute for Advanced Study of Mathematical Sciences, 4-21-1 Nakano, Nakano-ku, Tokyo, Japan
4
Dipartimento di Matematica, Politecnico di Milano, 20133 Milan, Italy 0123456789().: V,-vol
123
281
Page 2 of 23
V. A. Khoa et al.
Keywords Microscopic problems · Linearization · Well-posedness · Homogenization · Error estimates · Perforated domains Mathematics Subject Classification 35B27 · 35C20 · 35J91
1 Introduction 1.1 Microscopic problem Let Ω ε be a Lipschitz perforated domain contained in a polygonal bounded domain Ω ⊂ Rd (d = 2, 3). In this sense, Ω ε possesses a uniformly periodic microstructure defined by a length scale ε. This ε is a small parameter (0 < ε 1) since the size of the pores are usually much smaller than the characteristic length of the reservoir. We are herein concerned with the asymptotic behavior in a stationary case of the function u ε : Ω ε → R that describes the spread of concentration of solutes dissolved in a saturated porous tissue shaped by the perforated domain Ω ε with a cubic periodicity cell Y = [0, 1]d
Data Loading...
