Anizotropní hp-zjemňování v metoda virtuálních prvků
| Thesis title in Czech: | Anizotropní hp-zjemňování v metoda virtuálních prvků |
|---|---|
| Thesis title in English: | Anisotropic hp-refinement for the virtual element method |
| Key words: | Diferenciální rovnice|metoda virtuálních prvků|Voronoiovy buňky|polygonální prvky|numerické metody |
| English key words: | Differential equations|virtual element method|Voronoi cells|polygonal elements|numerical methods |
| Academic year of topic announcement: | 2025/2026 |
| Thesis type: | diploma thesis |
| Thesis language: | |
| Department: | Department of Numerical Mathematics (32-KNM) |
| Supervisor: | Scott Congreve, Ph.D. |
| Author: |
| Guidelines |
| This work will require study of the virtual element method [1] (VEM) and its hp-version [2-4] for
a singularly perturbed PDE. Then the application of the anisotropic $hp$-adaptive mesh refinement [5] to this problem will studied, and compared to the usual $hp$-adaptive algorithm [6] used. Additionally, the extension of this algorithm to meshes constructed from Voronoi diagrams will may be considered; note that a Voronoi mesh is the dual-mesh for a triangular mesh. The thesis will require writing code for validation in a C++-based polygonal VEM code [7]. |
| References |
| [1] Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., & Russo, A. Basic principles of virtual element methods. Math. Models. Methods. Appl. Sci. 23, 199–214 (2013).
[2] Mascotto, L. The hp version of the Virtual Element Method. PhD thesis, Università degli Studi di Milano, 2017. https://air.unimi.it/bitstream/2434/545778/2/phd_unimi_R10871.pdf [3] Beirão da Veiga, L., Chernov, A., Mascotto, L., & Russo, A. Exponential convergence of the hp virtual element method in presence of corner singularities. Numer. Math. 138, 581–613 (2018). https://doi.org/10.1007/s00211-017-0921-7 [4] Beirão da Veiga, L., Manzini, G. & Mascotto, L. A posteriori error estimation and adaptivity in hp virtual elements. Numer. Math. 143, 139–175 (2019). https://doi.org/10.1007/s00211-019-01054-6 [5] Dolejší, V. Anisotropic hp-adaptive method based on interpolation error estimates in the Lq-norm, App. Numer. Math. 82, 80-114 (2014). https://doi.org/10.1016/j.apnum.2014.03.003 [6] Melenk, J. M. & Wohlmuth, B. I., On residual-based a posteriori error estimation in $hp$-FEM. Adv. Comp. Math. 15, 311-331 (2001). http://dx.doi.org/10.1023/A:1014268310921 [7] PTEMS - PolyTopic Element Method Solver. https://bitbucket.org/congreves/ptems/src/master/ [8] Dedner, A., and Hodson, A. A framework for implementing general virtual element spaces. SIAM J. Sci. Comput. 46, 3 (2024), B229–B253. https://doi.org/10.1137/23M1573653 |
| Preliminary scope of work |
| The virtual element method (VEM) was first introduced in 2012 as an extended and generalised version of
the finite element and mimetic finite difference methods for second-order elliptic problems. The method is highly advantageous for many reasons including the ease with which the method extends to general polygonal and polyhedral meshes. This is extremely beneficial especially when developing adaptive schemes, since hanging nodes are automatically permissible within the VEM framework. Some work exists for the extension of the VEM to variable approximation order (polynomial degree) with isotropic adaptive refinement with the choice whether to perform mesh or polynomial refinement based based on comparing the convergence to an estimated convergence based on previous refinements. Other alternative exist, such as remeshing a triangle mesh for the whole domain to perform anisotropic refinement and variable polynomial degree based on minimising the interpolation error, which is advantage for problems containing boundary layers, such as the singularly perturbed problem, or internal interfaces. The aim of this thesis will be to apply this method to the VEM method for anisotropic refinement, and study whether it is possible to extend to a mesh based on a Voronoi diagram. |
| Preliminary scope of work in English |
| The virtual element method (VEM) was first introduced in 2012 as an extended and generalised version of
the finite element and mimetic finite difference methods for second-order elliptic problems. The method is highly advantageous for many reasons including the ease with which the method extends to general polygonal and polyhedral meshes. This is extremely beneficial especially when developing adaptive schemes, since hanging nodes are automatically permissible within the VEM framework. Some work exists for the extension of the VEM to variable approximation order (polynomial degree) with isotropic adaptive refinement with the choice whether to perform mesh or polynomial refinement based based on comparing the convergence to an estimated convergence based on previous refinements. Other alternative exist, such as remeshing a triangle mesh for the whole domain to perform anisotropic refinement and variable polynomial degree based on minimising the interpolation error, which is advantage for problems containing boundary layers, such as the singularly perturbed problem, or internal interfaces. The aim of this thesis will be to apply this method to the VEM method for anisotropic refinement, and study whether it is possible to extend to a mesh based on a Voronoi diagram. |