Doctoral theses
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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
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Περβολιανάκης, Χρήστος
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Thesis advisor
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Χατζηπαντελίδης Παναγιώτης
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Reviewer
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Μακριδάκης, Χαράλαμπος
Πλεξουσάκης Μιχαήλ
Χρυσαφίνος, Κωνσταντίνος
Κατσαούνης, Θεόδωρος
Μιτσούδης, Δημήτριος
Ζουράρης , Γεώργιος
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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.
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Language |
English |
Subject |
A posteriori error analysis |
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Elliptic reconstuction |
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Error analysis |
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Finite element method |
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Nonlinear parabolic problem |
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Positivity preservation |
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Ανάλυση σφάλματος |
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Διατήρηση θετικότητας |
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Εκ των υστέρων εκτίμηση |
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Μέθοδος πεπερασμένων στοιχείων |
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Μη -γραμμικό παραβολικό πρόβλημα |
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Χημειοταξία |
Issue date |
2023-03-17 |
Collection
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School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Doctoral theses
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Type of Work--Doctoral theses
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Permanent Link |
https://elocus.lib.uoc.gr//dlib/3/f/a/metadata-dlib-1679562001-207260-11592.tkl
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Views |
933 |
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