We prove the discrete compactness property of the edge elements of any order on a class of anisotropically refined meshes on polyhedral domains. The meshes, made up of tetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl. Sci. 21 (1998) 519–549]. They are appropriately graded near singular corners and edges of the polyhedron.

We prove the discrete compactness property of the edge elements of any order on a class of anisotropically refined meshes on polyhedral domains. The meshes, made up of tetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl. Sci. 21 (1998) 519–549]. They are appropriately graded near singular corners and edges of the polyhedron.

A reference triangular quadratic Lagrange finite element consists of a right triangle $\widehat{K}$ with unit legs $S_1$, $S_2$, a local space $\widehat{ℒ}$ of quadratic polynomials on $\widehat{K}$ and of parameters relating the values in the vertices and midpoints of sides of $\widehat{K}$ to every function from $\widehat{ℒ}$. Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping ${ℱ}_h=(F_1,F_2)\in \widehat{ℒ}\times \widehat{ℒ}$. We explicitly describe such invertible isoparametric mappings ${ℱ}_h$ for which the images ${ℱ}_h\left(S_1\right)$, ${ℱ}_h\left(S_2\right)$ of the segments $S_1$, $S_2$ are segments,...

In this paper we summarize three recent results in computational geometry, that were motivated by applications in mathematical modelling of fluids. The cornerstone of all three results is the genuine construction developed by D. Sommerville already in 1923. We show Sommerville tetrahedra can be effectively used as an underlying mesh with additional properties and also can help us prove a result on boundary-fitted meshes. Finally we demonstrate the universality of the Sommerville's construction by...

The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered...