Agglomeration and refinement of polytopic meshes for the virtual element method

• Název práce v češtině: Aglomerace a zjemňování polytopických sítích v metodě virtuálních prvků
• Název v anglickém jazyce: Agglomeration and refinement of polytopic meshes for the virtual element method
• Klíčová slova: Numerické řešení|odhady chyby|metoda virtuálních prvků|aglomerace
• Klíčová slova anglicky: Numerical solution|error estimates|virtual element method|agglomeration
• Akademický rok vypsání: 2023/2024
• Typ práce: disertační práce
• Jazyk práce: angličtina
• Ústav: Katedra numerické matematiky (32-KNM)
• Vedoucí / školitel: Scott Congreve, Ph.D.
• Řešitel: skrytý - zadáno a potvrzeno stud. odd.
• Datum přihlášení: 07.09.2023
• Datum zadání: 07.09.2023
• Datum potvrzení stud. oddělením: 03.10.2023

Zásady pro vypracování

Recently there has been significant work developing virtual element methods (VEM), which operate on meshes of general polytopic elements. Polytopal meshes can be defined directly, or constructed by agglomerating existing meshes of standard elements. Several methods for agglomeration already exists, which either attempt to maintain element geometry, or are based on optimal graph splitting techniques of the dual graph for the mesh. However, neither technique takes into account the results from the numerical analysis of the virtual element method. Therefore, in this thesis work we will develop an agglomeration technique based on metrics related to the analysis of the VEM, accounting for certain a priori knowledge; e.g. domain geometry details. Furthermore, we will also develop adaptive mesh refinement using these agglomeration techniques.

The thesis work will tackle the following tasks:
1. Development of metrics based on analysis of the VEM and construction of the agglomerated mesh based on optimisation of these metrics.
2. Validation and comparison to other agglomeration techniques.
3. Development of algorithm to allow for certain a priori information.
4. Adaptive refinement of VEM agglomerated meshes based on a posteriori error information.
5. Mesh agglomeration and refinement for time-dependent problems.

Further information can be found on the project website:
https://www2.karlin.mff.cuni.cz/~congreve/primus/

Seznam odborné literatury

A. Quarteroni, A. Valli: Numerical approximation of partial differential equations, Springer, 1997

P. F. Antonietti, L. Beirão da Veiga, G. Manzini: The Virtual Element Method and its Applications, Springer, 2022

L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, A. Russo: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

L. Beirão da Veiga, F. Brezzi, L. D. Marini, A. Russo The hitchhiker's guide to the virtual element method. Math. Models Methods Appl. Sci. 24(8) 1541–1573 (2014)

L. Beirão da Veiga, F. Brezzi, L. D. Marini, A. Russo: Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016)

L. Beirão da Veiga, F. Brezzi, L. D. Marini, A. Russo: Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM Math. Model. Numer. Anal. 50(3), 727–747 (2016)

L. Beirão da Veiga, C. Lovadina, A. Russo: Stability analysis for the virtual element method. Mathematical Models and Methods in Applied Sciences, 27(13):2557–2594 (2017)

A. Cangiani, G. Manzini, O.J. Sutton: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317–1354 (2017)

O. J. Sutton. Virtual Element Methods. PhD thesis, University of Leicester (2017)

G. Karypis, V. Kumar, V: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing, 20(1):359–392 (1999)

J. E. Jones, P.S. Vassilevski: AMGE based on element agglomeration. SIAM Journal on Scientific Computing, 23(1):109–133 (2001)

J. Kraus, J. Synka: An agglomeration-based multilevel-topology concept with application to 3D-FE meshes. Technical Report 2004-08, RICAM (2004)

S. Dargaville, A. Buchan, R. Smedley-Stevenson, P. Smith, and C. Pain,. A comparison of element agglomeration algorithms for unstructured geometric multigrid. Journal of Computational and Applied Mathematics, 390:113379 (2021)