Discrete Laplacians for general polygonal and polyhedral meshes
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Date
2024
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Abstract
This thesis presents several approaches that generalize the Laplace-Beltrami operator and its closely related gradient and divergence operators to arbitrary polygonal and polyhedral meshes.
We start by introducing the linear virtual refinement method, which provides a simple yet effective discretization of the Laplacian with the help of the Galerkin method from a Finite Element perspective.
Its flexibility allows us to explore alternative numerical schemes in this setting and to derive a second Laplacian, called the Diamond Laplacian with a similar approach, but this time combined with the Discrete Duality Finite Volume method.
It offers enhanced accuracy but comes at the cost of denser matrices and slightly longer solving times.
In the second part of the thesis, we extend the linear virtual refinement to higher-order discretizations. This method is called the quadratic virtual refinement method.
It introduces variational quadratic shape functions for arbitrary polygons and polyhedra. We also present a custom multigrid approach to address the computational challenges of higher-order discretizations,
making the faster convergence rates and higher accuracy of these polygon shape functions more affordable for the user.
The final part of this thesis focuses on the open degrees of freedom of the linear virtual refinement method.
By uncovering connections between our operator and the underlying tessellations, we can enhance the accuracy and stability of our initial method and improve its overall performance.
These connections equally allow us to define what a ``good'' polygon would be in the context of our Laplacian.
We present a smoothing approach that alters the shape of the polygons (while retaining the original surface as much as possible) to allow for even better performance.
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Keywords
Polygon meshes, Laplacians, Diefferntial operators, FEM