Agglomeration and refinement of polytopic meshes for the discontinuous Galerkin finite element method

• Thesis title in Czech: Aglomerace a zjemňování polytopických sítí v nespojité Galerkinově metodě
• Thesis title in English: Agglomeration and refinement of polytopic meshes for the discontinuous Galerkin finite element method
• Key words: Numerické řešení|odhady chyby|nespojitá Galerkinova metoda|aglomerace
• English key words: Numerical solution|error estimates|discontinuous Galerkin method|agglomeration
• Academic year of topic announcement: 2022/2023
• Thesis type: dissertation
• Thesis language: angličtina
• Department: Department of Numerical Mathematics (32-KNM)
• Supervisor: Scott Congreve, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 02.08.2022
• Date of assignment: 02.08.2022
• Confirmed by Study dept. on: 13.10.2022

Guidelines

Recently there has been significant work developing discontinuous Galerkin finite element methods (DGFEM) that 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 discontinuous Galerkin finite element method. Therefore, in this thesis work we will develop an agglomeration technique based on metrics related to the analysis of the DGFEM, 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 DGFEM 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 DGFEM 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/

References

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

V. Dolejsi, M. Feistauer: Discontinuous Galerkin Method - Analysis and Applications to Compressible Flow, Springer-Verlag, 2015

A. Cangiani, Z. Dong, E. H. Georgoulis P. Houston: hp-version Discontinuous Galerkin Methods of Polygonal and Polyhedral Meshes, Springer, 2017

A. Cangiani, E. H. Georgoulis, P. Houston: hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Model. Methods Appl. Sci. 24(10), 2009–2041 (2014)

S. Congreve, P. Houston: Two-grid hp-DGFEMs on agglomerated coarse meshes. PAMM, 19(1):e201900175 (2019)

S. Congreve, P. Houston: Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes. Technical Report, 2021. arXiv:2112.04540 [math.NA].

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)