Explicit Constants in Poincaré-Type Inequalities for Simplicial Domains and Application to A Posteriori Estimates

The paper is concerned with sharp estimates of constants in the classical Poincaré inequalities and Poincaré-type inequalities for functions with zero mean values in a simplicial domain or on a part of the boundary. These estimates are important for quantitative analysis of problems generated by differential equations where numerical approximations are typically constructed with the help of simplicial meshes. We suggest easily computable relations that provide sharp bounds of the respective constants and compare these results with analytical estimates (if such estimates are known). In the last section, we discuss possible applications and derive a computable majorant of the difference between the exact solution of a boundary value problem and an arbitrary finite dimensional approximation defined on a simplicial mesh.

A Dimension-Independent Representation for Multiresolution Nonmanifold Meshes

We consider the problem of representing and manipulating nonmanifold objects of any dimension and at multiple resolutions. We present a modeling scheme based on (1) a multiresolution representation, called the vertex-based nonmanifold multitessellation, (2) a compact and dimension-independent data structure, called the Simplified Incidence Graph (SIG), and (3) an atomic mesh update operator, called vertex-pair contraction/vertex expansion. We propose efficient algorithms for performing the vertex-pair contraction on a simplicial mesh encoded as a SIG, and an effective representation for encoding this multiresolution model based on a compact encoding of vertex-pair contractions and vertex expansions.

A COST/BENEFIT ANALYSIS OF SIMPLICIAL MESH IMPROVEMENT TECHNIQUES AS MEASURED BY SOLUTION EFFICIENCY

The quality of unstructured meshes has long been known to affect both the efficiency and the accuracy of the numerical solution of application problems. Mesh quality can often be improved through the use of algorithms based on local reconnection schemes, node smoothing, and adaptive refinement or coarsening. These methods typically incur a significant cost, and in this paper, we provide an analysis of the tradeoffs associated with the cost of mesh improvement in terms of solution efficiency. We first consider simple finite element applications and show the effect of increasing the number of poor quality elements in the mesh and decreasing their quality on the solution time of a number of different solvers. These simple application problems are theoretically well-understood, and we show the relationship between the quality of the mesh and the eigenvalue spectrum of the resulting linear system. We then consider realistic finite element and finite volume application problems, and show that the cost of mesh improvement is significantly less than the cost of solving the problem on a poorer quality mesh.