On nonobtuse refinements of tetrahedral finite element meshes

It is known that piecewise linear continuous finite element (FE) approximations on nonobtuse tetrahedral FE meshes guarantee the validity of discrete analogues of various maximum principles for a wide class of elliptic problems of the second order. Such analogues are often called discrete maximum principles (or DMPs in short). In this work we present several global and local refinement techniques which produce nonobtuse conforming (i.e. face-to-face) tetrahedral partitions of polyhedral domains. These techniques can be used in order to compute more accurate FE approximations (on finer and/or adapted tetrahedral meshes) still satisfying DMPs

Nonobtuse local tetrahedral refinements towards a polygonal face/interface

In this note we show how to generate and conformally refine nonobtuse tetrahedral meshes locally in the neighbourhood of a polygonal face or a polygonal interior interface of a three-dimensional domain. The technique proposed can be used for example for problems with boundary and/or interior layers, and for interface problems

Some discrete maximum principles arising for nonlinear elliptic finite element problems

The discrete maximum principle (DMP) is an important measure of the qualitative reliability of the applied numerical scheme for elliptic problems. This paper starts with formulating simple sufficient conditions for the matrix case and for nonlinear forms in Banach spaces. Then a DMP is derived for finite element solutions for certain nonlinear partial differential equations: we address nonlinear elliptic problems with mixed boundary conditions and interface conditions, allowing possibly degenerate nonlinearities and thus extending our previous results

On simplicial red refinement in three and higher dimensions

summary:We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one

Red refinements of simplices into congruent subsimplices

We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one

Local nonobtuse tetrahedral refinements around an edge

In this note we show how to generate and conformly refine nonobtuse tetrahedral meshes locally around and towards an edge so that all dihedral angles of all resulting tetrahedra remain nonobtuse. The proposed technique can be used e.g. for a numerical treatment of solution singularities, and also for various mesh adaptivity procedures, near the reentrant corners of cylindric-type 3D domains

On Conforming Tetrahedralisations of Prismatic Partitions

We present an algorithm for conform (face-to-face) subdividing prismatic partitions into tetrahedra. This algorithm can be used in the finite element calculations and analysis

On continuous and discrete maximum/minimum principles for reaction-diffusion problems with the Neumann boundary condition

summary:In this work, we present and discuss continuous and discrete maximum/minimum principles for reaction-diffusion problems with the Neumann boundary condition solved by the finite element and finite difference methods