Intensive lectures by Prof. Carstensen (2023.02.08~2023.02.17)
Primer on adaptive finite element methods
by Carsten Carstensen of Humboldt-Universität zu Berlin, Germany
https://yonsei.zoom.us/j/96399623909
회의 ID: 963 9962 3909
The short course delivers an introduction to adaptive mesh-refining algorithms for the spatial discretisation of partial differential equations. Based on striking numerical evidence, the main message is the wisdom that any finite element analysis in practise has to utilise an adaptive mesh-refining strategy to achieve optimal convergence rates. The primer provides a set of four abstract conditions, known as the axioms of adaptivity, that guarantee even optimal convergence rates in a clear sense of nonlinear approximation classes. The Poisson model problem represents a boundary value problem with more general elliptic second-order partial differential equations and marks the point of departure for with the simplest finite element method in 2D named after Courant and also described as conforming P1 finite elements. The data structures for the underlying triangulations will be explained as well as the mesh-refining algorithms that implement any (local or global) mesh-refinement. Besides ten lines of pseudo-code that realise a running MATLAB code for the assembling and solve of the approximation of the solution to the Poisson model problem, the newest-vertex algorithm and its realisation will be explained.
The practical aspects of the adaptive mesh-refining algorithms based on the D\"orfler marking are complemented by theoretical concepts required for the analysis of the optimal rates. This forms the platform of the axioms of adaptivity in an abstract format (so they are application-independent) that are sufficient for convergences and optimal rates for an estimator. The proofs of all that are somehow abstract but (relatively) simple. The point is to verify those in the application at hand. This will be examined for the Poisson model problem and the first-order nonconforming Crouzeix-Raviart finite element method for the Stokes equations in the finish of this primer. It is remarkable that the axioms apply to almost all successful applications of adaptive finite element schemes in some variation or generalisation as the basic mathematical argument.
Syllabus on the eight days
Wed. Feb. 8 / 14:00
Data and algorithms. Regular triangulations; Euler formulas for polygonal meshes; local, global and coefficient matrices, assembling; finite element program in less than 10 lines of MATLAB; counter example to saturation; conforming and non-conforming P1 finite elements; discrete Helmholtz decomposition
Thu. Feb. 9 / 14:00
A~priori error analysis. Galerkin orthogonality; Cea lemma, general convergence of Galerkin schemes; arbitrarily poor convergence rates; P1 finite element interpolation; maximal angle condition; divergence of the finite element method; minimal angle condition for inverse estimates; graded meshes; numerical benchmarks for improved convergence rates
Fri. Feb. 10 / 14:00
Adaptive mesh-refining. One-level refinements; algorithms for bisection; closure algorithms for refinements;routine refine; the refine lemma; newest-vertex bisection; implementation of newest-vertex bisection; uniform shape regularity.
Mon. Feb. 13 / 14:00
Admissible triangulations. Definition by successive one-level refinements; forest of binary trees; finer relation; characterisation; overlay of triangulations; minimality of routine refine; overhead control theorem by Binev-Dahmen-DeVore.
Tue. Feb. 14 / 14:00
Axioms of adaptivity imply convergence. Stability; reduction; discrete reliability; quasi-orthogonality; estimator convergence; R-linear convergence.
Wed. Feb. 15 / 14:00
Axioms of adaptivity imply optimal rates. Optimality in terms of nonlinear approximation classes; comparison lemma; proof of optimal convergence rates for adaptive mesh-refining algorithms.
Thu. Feb. 16 / 14:00
Optimal convergence rates for adaptive P1 conforming finite element methods in Poisson model problem. Discrete trace inequality, jump control, proof of stability and reduction, discrete quasi-interpolation, discrete reliability.
Fri. Feb. 17 / 14:00
Optimal convergence rates for adaptive P1 nonconforming finite element methods in Stokes equations. Implementation of Crouzeix-Raviart finite elements; discrete Helmholtz decomposition; generalised quasi-orthogonality; numerical experiments.
- Selected references:
C. Carstensen: Yonsei Lectures on the Finite Element Method. Department Computational Science and Engineering, Yonsei University, Seoul, Korea; provided at the institute.
C. Carstensen, M. Feischl, M. Page, D. Praetorius: Axioms of adaptivity, Comput. Math. Appl., 67, pp. 1195–1253, 2014; open access.
C. Carstensen, E.-J. Park: Convergence and optimality of adaptive least squares finite element methods, SIAM J. Numer. Anal., 53, pp. 43–62, 2015.
C. Carstensen, H. Rabus: Axioms of adaptivity with separate marking for data resolution, SIAM J. Numer. Anal., 55, pp. 2644–2665, 2017.
- Organizer: Eun-Jae Park (ejpark@yonsei.ac.kr)