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 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

Adaptive finite element method assisted by stochastic simulation of chemical systems

Stochastic models of chemical systems are often analysed by solving the corresponding\ud Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density

Tetrahedral Meshes in Biomedical Applications: Generation, Boundary Recovery and Quality Enhancements

Mesh generation is a fundamental precursor to finite element implementations for solution of partial differential equations in engineering and science. This dissertation advances the field in three distinct but coupled areas. A robust and fast three dimensional mesh generator for arbitrarily shaped geometries was developed. It deploys nodes throughout the domain based upon user-specified mesh density requirements. The system is integer and pixel based which eliminates round off errors, substantial memory requirements and cpu intensive calculations. Linked, but fully detachable, to the mesh generation system is a physical boundary recovery routine. Frequently, the original boundary topology is required for specific boundary condition applications or multiple material constraints. Historically, this boundary preservation was not available. An algorithm was developed, refined and optimized that recovers the original boundaries, internal and external, with fidelity. Finally, a node repositioning algorithm was developed that maximizes the minimum solid angle of tetrahedral meshes. The highly coveted 2D Delaunay property that maximizes the minimum interior angle of a triangle mesh does not extend to its 3D counterpart, to maximize the minimum solid angle of a tetrahedron mesh. As a consequence, 3D Delaunay created meshes have unacceptable sliver tetrahedral elements albeit composed of 4 high quality triangle sides. These compromised elements are virtually unavoidable and can foil an otherwise intact mesh. The numerical optimization routine developed takes any preexisting tetrahedral mesh and repositions the nodes without changing the mesh topology so that the minimum solid angle of the tetrahedrons is maximized. The overall quality enhancement of the volume mesh might be small, depending upon the initial mesh. However, highly distorted elements that create ill-conditioned global matrices and foil a finite element solver are enhanced significantly

Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space V_{H,P}. The resulting `coarse' numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space V_{h,p}; thereby, only a linear system of equations is solved on the richer space V_{h,p}. In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces V_{H,P} and V_{h,p}, respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the cpu time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method

Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space V(TH,P). The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space V(Th,p); thereby, only a linear system of equations is solved on the richer space V(Th,p). In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces V(TH,P) and V(Th,p), respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the CPU time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method