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Identifier 000454456
Title Numerical methods for evolutionary differential equations that maintain the positivity of the solution
Alternative Title Αριθμητικές μέθοδοι για δυναμικές διαφορικές εξισώσεις που διατηρούν την θετικότητα της λύσης
Author Περβολιανάκης, Χρήστος
Thesis advisor Χατζηπαντελίδης Παναγιώτης
Reviewer Μακριδάκης, Χαράλαμπος
Πλεξουσάκης Μιχαήλ
Χρυσαφίνος, Κωνσταντίνος
Κατσαούνης, Θεόδωρος
Μιτσούδης, Δημήτριος
Ζουράρης , Γεώργιος
Abstract We consider the Keller-Segel model of chemotaxis on a bounded domain Ω - R2. Assuming appropriate conditions in order to have bounded solutions for all times, we discretize the spatial variable with the standard linear finite element method and the temporal variable with implicit Euler and the implicit midpoint rule. We prove that the resulting semidiscrete and fully-discrete scheme via backward Euler can take negative values, therefore our aim is to seek conditions on existing or appropriate modifications of finite element based discretization methods that preserve positivity of the system solution. We recall the results conserving the positivity and the mass conservation of the stabilized schemes which are introduced in [65]. Under suitable smoothness assumptions on the solution, we prove the existence and the uniqueness of the solution of these stabilized schemes and its solution remains positive under a suitable choice of mesh size and time step. Moreover, we prove results concerning the mass conservation and we prove error estimates in L2 and H1-norm in space and L∞ in time. Numerical experiments for various test problems are performed in order to study the asymptotic behavior of the error in the L2 and H1-norm in space and L∞ in time. Besides the a priori analysis on the stabilized schemes of [65], we derive residual-based a posteriori error estimates for the fully discrete schemes of implicit Euler and implicit midpoint rule. Since we have a parabolic system of partial differential equations, the analysis includes the appropriate introduction of an elliptic reconstruction operator, similar to [50]. The space discretization uses the standard linear finite element spaces and the time discretization is based on the implicit Euler and in implicit midpoint rule. We also derive a posteriori error estimates in L∞(L2) and L2(H1)-norm. Numerical experiments for various test problems are performed in order to study the asymptotic behavior of the a posteriori error estimators.
Language English
Subject A posteriori error analysis
Elliptic reconstuction
Error analysis
Finite element method
Nonlinear parabolic problem
Positivity preservation
Ανάλυση σφάλματος
Διατήρηση θετικότητας
Εκ των υστέρων εκτίμηση
Μέθοδος πεπερασμένων στοιχείων
Μη -γραμμικό παραβολικό πρόβλημα
Χημειοταξία
Issue date 2023-03-17
Collection   School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Doctoral theses
  Type of Work--Doctoral theses
Permanent Link https://elocus.lib.uoc.gr//dlib/3/f/a/metadata-dlib-1679562001-207260-11592.tkl Bookmark and Share
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