Galerkin method and implementation of functional type a posteriori error estimates with black-box solvers for Linear Elasticity

• 主讲人：Maksim Frolov (Peter the Great St. Petersburg Polytechnic University)
• 举办方： Beijing-Saint Petersburg Mathematics Colloquium
• 时间： 2025-04-24 21:00 - 2025-04-24 22:00
• 地点： Online

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Abstract: A posteriori error estimates provide an explicit accuracy control for numerical solutions of boundary-value problems for Partial Differential Equations.

It is an important and practically significant area of research in applied mathematics. Regardless of the fact that the classical methods of a posteriori error control are investigated very intensively for several decades, functional-type a posteriori estimates are very promising. This approach is fully reliable and it can be used to solvers with some hidden details of numerical implementations. Such estimates are known for many problems of the elasticity theory. However, as follows from the work of Prof. S. Repin and Dr. A. Muzalevsky, when implementing a posteriori estimates of the functional type, the use of classical approximations of the Finite Element Method leads to a growing overestimation of the absolute value of the error. Later, in the work of M. Frolov, this effect is highlighted more transparently. It is shown that the use of approximations that are more natural for mixed finite element methods avoids a growing overestimation of the absolute error value with mesh refinements. Comprehensive numerical testing and justification of this approach are provided in joint papers with Dr. M. Churilova and Ph.D. student D. Petukhov. In particular, a comparative analysis is performed for zero-order and first-order Raviart-Thomas finite elements implemented in MATLAB by D. Petukhov. For plane problems of linear elasticity, it is shown that the use of the first-order Raviart-Thomas approximation significantly reduces an overestimation of the absolute error value.

 

Bio: Maksim Evgenyevich Frolov has received his Ph.D. in Applied Mathematics in 2004 (thesis “A posteriori error estimates for approximate solutions to variational problems for elliptic equations of divergent type”) and D.Sci. in 2015 (with work “Functional methods and its implementation to a posteriori error control in Linear Elasticity”). From 2016 to 2024 he was the Director of the Institute of Physics and Mechanics at Peter the Great St. Petersburg Polytechnic University (the historical name PhysMech was returned in 2021; for the period 2013-2021 — the Institute of Applied Mathematics and Mechanics). At present he is a professor of this institute.

His research interests include reliable modelling, a posteriori error estimates, numerical methods for PDE’s (Finite Element Methods), computational mechanics. Prof. Frolov constructed new reliable error estimates for several practically important models in Linear Elasticity theory: the biharmonic equation, Timoshenko beams, Reissner-Mindlin plates (joint work with Prof. S. Repin from PDMI RAS, Russia and Prof. P. Neittaanmäki from University of Jyväskylä, Finland), the Cosserat theory of elasticity in 2D (joint work with Prof. S. Repin) and 3D. He also introduced new methods of implementation of the functional approach to a posteriori error control in adaptive algorithms in 2D based on Raviart-Thomas and Arnold-Boffi-Falk approximations.