Interpolation and scattered data fitting on manifolds using projected Powell–Sabin splines

We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(Uξ , ξ)}ξ∈ satisfying certain conditions of smooth dependence on ξ. If is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function

Parametric Interpolation To Scattered Data [QA281. A995 2008 f rb].

Dua skema interpolasi berparameter yang mengandungi interpolasi global untuk data tersebar am dan interpolasi pengekalan-kepositifan setempat data tersebar positif dibincangkan. Two schemes of parametric interpolation consisting of a global scheme to interpolate general scattered data and a local positivity-preserving scheme to interpolate positive scattered data are described

B-spline-like bases for C2C^2 cubics on the Powell-Sabin 12-split

For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a B\'ezier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for $C^0$-, $C^1$-, and $C^2$-smoothness are derived

Macro-element interpolation on tensor product meshes

A general theory for obtaining anisotropic interpolation error estimates for macro-element interpolation is developed revealing general construction principles. We apply this theory to interpolation operators on a macro type of biquadratic $C^1$ finite elements on rectangle grids which can be viewed as a rectangular version of the $C^1$ Powell-Sabin element. This theory also shows how interpolation on the Bogner-Fox-Schmidt finite element space (or higher order generalizations) can be analyzed in a unified framework. Moreover we discuss a modification of Scott-Zhang type giving optimal error estimates under the regularity required without imposing quasi uniformity on the family of macro-element meshes used. We introduce and analyze an anisotropic macro-element interpolation operator, which is the tensor product of one-dimensional $C^1-P_2$ macro interpolation and $P_2$ Lagrange interpolation. These results are used to approximate the solution of a singularly perturbed reaction-diffusion problem on a Shishkin mesh that features highly anisotropic elements. Hereby we obtain an approximation whose normal derivative is continuous along certain edges of the mesh, enabling a more sophisticated analysis of a continuous interior penalty method in another paper

An adaptive octree finite element method for PDEs posed on surfaces

The paper develops a finite element method for partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method uses traces of bulk finite element functions on a surface embedded in a volumetric domain. The bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures. The cartesian structure of the bulk mesh leads to easy and efficient adaptation process, while the trace finite element method makes fitting the mesh to the surface unnecessary. The number of degrees of freedom involved in computations is consistent with the two-dimension nature of surface PDEs. No parametrization of the surface is required; it can be given implicitly by a level set function. In practice, a variant of the marching cubes method is used to recover the surface with the second order accuracy. We prove the optimal order of accuracy for the trace finite element method in $H^1$ and $L^2$ surface norms for a problem with smooth solution and quasi-uniform mesh refinement. Experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom, if grid adaptation is based on an appropriate error indicator. The paper shows results of numerical experiments for a variety of geometries and problems, including advection-diffusion equations on surfaces. Analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted (trace) finite elements provide simple, efficient, and reliable tool for numerical treatment of PDEs posed on surfaces

Planewave density interpolation methods for 3D Helmholtz boundary integral equations

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nystr\"om and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nystr\"om and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique