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The energy of the Francfort-Marigo model of brittle fracture can be approximated, in the sense of G-convergence, by the Ambrosio-Tortorelli functional. In this work, we formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers. For each algorithm, we combine a Newton-type method with residual-driven adaptive mesh refinement. We present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the Ambrosio-Tortorelli functional. [ABSTRACT FROM AUTHOR]

We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulations of domains in ℝd are orthogonal. Compared with linear interpolation, the resulting approximation schemes yield superior accuracy at little extra cost. [ABSTRACT FROM AUTHOR]

We establish a general framework for analyzing the class of finite volume methods for the Stokes equations. Under the framework, optimal order convergence for velocity and pressure in various Sobolev norms are obtained in a natural and systematic way. Specifically, optimal L² error estimates for velocity are obtained for the first time for several different finite volume methods. [ABSTRACT FROM AUTHOR]

A rational approximation in a triangle is proposed and analyzed in this paper. The rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform. Optimal error estimates for the L2- and H01 -orthogonal projections are derived with upper bounds expressed in the original coordinates in the triangle. It is shown that the rational approximation is as accurate as the polynomial approximation in the triangle. Illustrative numerical results, which are in agreement with the theoretical estimates, are presented. [ABSTRACT FROM AUTHOR]

We introduce a new finite element method, the composite mini element, for the mixed discretization of the Stokes equations on two- and three-dimensional domains that may contain a huge number of geometric details. In standard finite element discretizations of the Stokes problem, such as the classical mini element, the approximation quality is determined by the maximal mesh size of the underlying triangulation, while the computational effort is determined by its number of elements. If the physical domain is very complicated, then the minimal number of simplices, which are necessary to resolve the domain, can be very large and distributed in a nonoptimal way with respect to the approximation quality. In contrast to that, the minimal dimension of the composite mini element space is independent of the number of geometric details. Instead of a geometric resolution of the domain and the boundary condition by the finite element mesh the shape of the finite element functions is adapted to the geometric details. This approach allows low-dimensional approximations even for problems with complicated geometric details such as holes or rough boundaries. We prove its linear (optimal order) approximability and its inf-sup stability. Further, we will be able to control the nonconformity in the space without increasing the space dimension in such a way that the a priori error estimate ∥ u - uCME∥, 1,Ω + ∥p - pCME∥0,Ω ≲ h ∥ f ∥ 0,Ω holds. Thereby, in contrast to the classical methods, the choice of the mesh size parameter h is not constrained by the size of geometric details. In addition, it turns out that the method can be viewed as a coarse-scale generalization of the classical mini element approach; i.e., it reduces the computational effort, while the approximation quality depends on the (coarse) mesh size in the usual way. [ABSTRACT FROM AUTHOR]

In this paper, we develop a postprocessing derivative recovery scheme for the finite element solution uh on general unstructured but shape regular triangulations. In the case of continuous piecewise polynomials of degree p ≥ 1, by applying the global L² projection (Qh) and a smoothing operator (Sh), the recovered pth derivatives (Shm Qh ∂p uh) superconverge to the exact derivatives (∂pu). Based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the (p+1)st derivatives approximated by ∂Shm Qh∂puh. We provide several numerical examples illustrating the effectiveness of our schemes. We also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence. [ABSTRACT FROM AUTHOR]

In Part I of this work, we develop superconvergence estimates for piecewise linear finite element approximations on quasi-uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In particular, we first show a superconvergence of the gradient of the finite element solution uh and to the gradient of the interpolant uI. We then analyze a postprocessing gradient recovery scheme, showing that Qh∇uh is a superconvergent approximation to ∇u. Here Qh is the global L2 projection. In Part II, we analyze a superconvergent gradient recovery scheme for general unstructured, shape regular triangulations. This is the foundation for an a posteriori error estimate and local error indicators. [ABSTRACT FROM AUTHOR]

Presents algorithms for computing C[sup 1] quartic interpolating splines. Availability of full approximation power with approximation constants; Review of the concept of minimal determining sets; Introduction of type-O triangulations.

We describe a numerical method to construct volume matching C¹ splines satisfying the frequently occurring requirement of nonnegativity. The approximated histogram is assumed to be defined on a partition of a polygonal domain into polygonal subdomains. Our idea consists of applying cubic C&sub1; splines on Clough-Tocher refinements of a triangulation that is compatible with the given partition. The proposed algorithm always works successfully for nonnegative histograms. [ABSTRACT FROM AUTHOR]

This paper proposes a weak patch test that improves both the patch test and the generalized patch test. Under a weak superapproximation assumption, which can be obtained from certain approximation and weak continuity property, it is shown that the weak patch test is equivalent to the generalized patch test. Thus if a nonconforming finite element passes the patch test and satisfies some approximation and weak continuity property, its convergence is then guaranteed. Furthermore, the consistency term of such a nonconforming element can be proved to be of order O(h). It is shown that if a nonconforming element does not pass the patch test and a condition relative to subdividing a domain by the element is true, then one can find a family of triangulations for which the element is divergent. [ABSTRACT FROM AUTHOR]

In this paper we establish linear independence and stability of certain piecewise linear prewavelets over arbitrary bounded triangulations. These prewavelets are natural generalizations of the locally supported element constructed by Kotyczka and Oswald for an infinite three-directional mesh. [ABSTRACT FROM AUTHOR]