103 research outputs found
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
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 modifications of continuous and discrete maximum principles for reaction-diffusion problems
In this work, we present and discuss some modifications, in the form of two-sided estimation (and also for arbitrary source functions instead of usual sign-conditions), of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods
Discrete maximum principles for nonlinear parabolic PDE systems
Discrete maximum principles (DMPs) are established for finite element approximations of systems of nonlinear parabolic partial differential equations with mixed boundary and interface conditions. The results are based on an algebraic DMP for suitable systems of ordinary differential equations
On the maximum angle condition for the conforming longest-edge n-section algorithm for large values of n
In this note we introduce the conforming longest-edge -section algorithm and show that for it produces a family of triangulations which does not satisfy the maximum angle condition
A posteriori error analysis of a stabilized mixed FEM for convectuion-diffusion problems
We present an augmented dual-mixed variational formulation for a linear convection-diffusion equation with homogeneous Dirichlet boundary conditions. The approach is based on the addition of suitable least squares type terms. We prove that for appropriate values of the stabilization parameters, that depend on the diffusion coefficient and the magnitude of the convective velocity, the new variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds. In particular, we derive the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart-Thomas and continuous piecewise polynomials. In addition, we introduce a simple a posteriori error estimator which is reliable and locally efficient. Finally, we provide numerical experiments that illustrate the behavior of the method
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