We consider the topic of partitioning unstructured finite element meshes by a class of multilevel graph partitioning algorithms. Two issues are studied, where the first issue concerns the coarsening phase in such multilevel graph partitioning algorithms. In particular, we propose a new heuristic for matching the vertices of a graph during the coarsening phase. We compare our heuristic with two other known matching heuristics in respect of matching ratio and quality of the final partition. As the second issue of the paper, we look at the relation between the parallel effciency of finite element computation and different aspects of the partition quality.