Search

In this work, we present an adaptive unfitted finite element scheme that combines the aggregated finite element method with parallel adaptive mesh refinement. We introduce a novel scalable distributed-memory implementation of the resulting scheme on locally adapted Cartesian forest-of-trees meshes. We propose a two-step algorithm to construct the finite element space at hand by means of a discrete extension operator that carefully mixes aggregation constraints of problematic degrees of freedom, which get rid of the small cut cell problem, and standard hanging degree of freedom constraints, which ensure trace continuity on nonconforming meshes. Following this approach, we derive a finite element space that can be expressed as the original one plus well-defined linear constraints. Moreover, it requires minimum parallelization effort, using standard functionality available in existing large-scale finite element codes. Numerical experiments demonstrate its optimal mesh adaptation capability, robustness to cut location, and parallel efficiency, on classical Poisson hp-adaptivity benchmarks. Our work opens the path to functional and geometrical error-driven dynamic mesh adaptation with the aggregated finite element method in large-scale realistic scenarios. Likewise, it can offer guidance for bridging other scalable unfitted methods and parallel adaptive mesh refinement. [ABSTRACT FROM AUTHOR]

The computation of eigenvalues of large-scale matrices arising from finite element discretizations has gained significant interest in the last decade (Knyazev et al. in Numerical solution of PDE eigenvalue problems, vol 56. Mathematisches Forschungsinstitut, Oberwolfach, ). Here we present an new algorithm based on slicing the spectrum that takes advantage of the rank structure of resolvent matrices in order to compute $$m$$ eigenvalues of the generalized symmetric eigenvalue problem in $${\fancyscript{O}}(nm\log ^\alpha n)$$ operations, where $$\alpha >0$$ is a small constant. [ABSTRACT FROM AUTHOR]

It has been shown that combinatorial optimization of matrix-vector multiplication can lead to faster evaluation of finite element stiffness matrices. Based on a graph model characterizing relationships between rows, an efficient set of operations can be generated to perform matrix-vector multiplication for this problem. We improve the graph model by extending the set of binary row relationships and solve this combinatorial optimization problem optimally for the binary row relationships implemented, yielding significantly improved results over previously published graph models. We also extend the representation by using hypergraphs to model more complicated row relationships, expressing a three-row relationship with a three-vertex hyperedge, for example. Our initial greedy algorithm for this hypergraph model has yielded significantly better results than the graph model for many matrices. [ABSTRACT FROM AUTHOR]