In Part I of this work, we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes satisfying an O(h2) approximate parallelogram property. In this work, we consider superconvergence for general unstructured but shape regular meshes. We develop a postprocessing gradient recovery scheme for the finite element solution uh, inspired in part by the smoothing...

A dynamic adaptive numerical method for solving partial differential equations on the sphere is developed. The method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids, and is an extension of the adaptive wavelet collocation method to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. An OðN Þ hierarchical fini...

Mesh reduction techniques are used for accelerating the visualization process for large datasets. Typical examples are scalar or vector valued functions deened on complex 2 or 3 dimensional meshes. Grosso et al. presented a method for mesh optimization based on nite elements approximations with the L2 norm and adaptive local mesh reenement. Starting with a very coarse triangulation of the funct...

In this article, we propose a Partition ofUnity Refinement (PUR)method to improve the local approximations of elliptic boundary value problems in regions of interest. The PUR method only needs to refine the local meshes and hanging nodes generate no difficulty. The mesh qualities such as uniformity or quasi-uniformity are kept. The advantages of the PUR include its effectiveness and relatively ...

An improved method for adaptively constructing a terrain surface representation from a set of data points is presented. Refinement and decimation steps are repeatedly applied to triangular meshes, incrementally determining a better distribution of the data points, while a specified error tolerance is preserved. Even though not asymptotically optimal or monotonically convergent, it produces appr...

Several existing algorithms for computing the matrix cosine employ polynomial or rational approximations combined with scaling and use of a double angle formula. Their derivations are based on forward error bounds. We derive new algorithms for computing the matrix cosine, the matrix sine, and both simultaneously, that are backward stable in exact arithmetic and behave in a forward stable manner...

We present new MUSCL techniques associated with cell-centered Finite Volume method on triangular meshes. The first reconstruction consists in calculating a one vectorial slope per control volume by a minimization procedure with respect to a prescribed stability condition. The second technique we propose is based on the computation of three scalar slopes per triangle (one per edges) still respec...

A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein–Bézier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The o...

The phase-field-crystal model for liquid crystals is solved numerically in two spatial dimensions. This model is formulated with three position-dependent order parameters, namely the reduced translational density, the local nematic order parameter, and the mean local direction of the orientations. The equilibrium free-energy functional involves local powers of the order parameters up to fourth ...

We present a family of mixed methods for linear elasticity, that yield exactly symmetric, but only weakly conforming, stress approximations. The method is presented in both two and three dimensions (on triangular and tetrahedral meshes). The method is efficiently implementable by hybridization. The degrees of freedom of the Lagrange multipliers, which approximate the displacements at the faces,...