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The skeletal finite element method (FEM) in this paper is a hybridized discontinuous Galerkin FEM with the Lehrenfeld--Sch\"oberl stabilization and also known as a weak Galerkin FEM. With an appropriate stabilization, it provides eigenvalue approximations for the Laplacian on any regular triangulation \scrT with maximal mesh-size hmax, which are guaranteed lower eigenvalue bounds (GLB) if they are sufficiently large. This paper establishes a bound \alpha (\scrT) for a global stabilization parameter \alpha such that \alpha \leq \alpha (\scrT) leads to an eigenvalue approximation \lambda h \leq \lambda for the exact eigenvalue \lambda, provided \kappa 2 CRh2 max \lambda h \leq 1 for a universal constant \kappa CR. For a 2D triangulation \scrT into triangles, a comparison with the bound CRGLB := \lambda CR/(1 + \varepsilon \lambda CR) \leq \lambda from [C. Carstensen and J. Gedicke, Math. Comp., 83 (2014), pp. 2605--2629] proves under the same conditions that CRGLB \leq \lambda h \leq \lambda. The paper also provides an alternative proof of the already established asymptotic lower bound property. [ABSTRACT FROM AUTHOR]

Mixed finite element methods with ux errors in H(div)-norms and div-least-squares finite element methods require a separate marking strategy in obligatory adaptive mesh-refining. The refinement indicator Ψ2(T;K) = n²(T;K) + µ²(K) of a finite element domain K in an admissible triangulation T consists of some residual-based error estimator n(T;K) with some reduction property under local mesh-refining and some data approximation error µ(K). Separate marking means either Dorer marking if µ²(T) =n²(T) or otherwise an optimal data approximation algorithm with controlled accuracy. The axioms are sucient conditions on the estimators n(T;K) and data approximation errors µ(K) for optimal asymptotic convergence rates. The enfolded set of axioms of this paper simplifies [C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Comput. Math. Appl., 67 (2014), pp. 1195{1253] for collective marking, treats separate marking established for the first time in an abstract framework, generalizes [C. Carstensen and E.-J. Park, SIAM J. Numer. Anal., 53 (2015), pp. 43{62] for least-squares schemes, and extends [C. Carstensen and H. Rabus, Math. Comp., 80 (2011), pp. 649{667] to the mixed finite element method with ux error control in H(div). The paper gives an outline of the mathematical analysis for optimal convergence rates but also serves as a reference so that future contributions merely verify a few axioms in a new application in order to ensure optimal mesh-refinement of the adaptive algorithm. [ABSTRACT FROM AUTHOR]

A family of algebraic flux correction (AFC) schemes for linear boundary value problems in any space dimension is studied. These methods' main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection-diffusion-reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness. [ABSTRACT FROM AUTHOR]

The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted to this sphere is a quality measure for volume elements. We show that the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra, hexahedra, prisms, pyramids, and octahedra, that is, the optimization process converges to regular polyhedra. We show that the (normalized) gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method. Furthermore, we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally. [ABSTRACT FROM AUTHOR]

We consider the coupling of the finite volume method and the boundary element method of Erath [SIAM J. Numer. Anal., 50 (2012), pp. 574-594] in two and three dimensions. This method can be used, for example, to approximate a solution of the transport of a concentration in a fluid, where no boundary conditions are available. We derive residual-based a posteriori estimates (also for an upwind version). These upper and lower bounds measure the error in an energy (semi)norm between the exact solution and the numerical solution. The upper bound is robust in the sense that it does not depend on the variation of the model data. The lower bound, however, depends additionally on the local Péclet number. The local contributions of the a posteriori estimates are used to steer an adaptive mesh-refining algorithm. This strategy turns out to be very suitable for the numerical treatment of transmission problems, which have singularities or boundary/internal layers. Several numerical examples illustrate the effectiveness of the new conservative adaptive coupling method. [ABSTRACT FROM AUTHOR]

We prove that a planar C²-regular boundary Γ can always be parameterized with its closest point projection π over a certain collection of edges Γh in an ambient triangulation by making simple assumptions on the background mesh. For Γh, we select the edges that have both vertices on one side of Γ and belong to a triangle that has a vertex on the other side. By imposing restrictions on the size of triangles near the curve and by requesting that certain angles in the mesh be strictly acute, we prove that π: Γh → Γ is a homeomorphism and that it is C¹ on each edge in Γh, and we provide bounds for the Jacobian of the parameterization. The assumptions on the background mesh are both easy to satisfy in practice and conveniently verified in computer implementations. The parameterization analyzed here was previously proposed by the authors and applied to the construction of high-order curved finite elements on a class of planar piecewise C²-curves. [ABSTRACT FROM AUTHOR]

In this paper a Nitsche-type discretization based on a discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region Ω which is a union of N disjoint polygonal subdomains Ω i of diameter O(Hi). The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces ∂Ωi. Inside each subdomain, a conforming finite element space on a quasi-uniform triangulation with mesh size O(hi) is introduced. To handle the nonmatching meshes across the subdomain interfaces, a DG discretization is applied only on the interfaces. For solving the resulting discrete system, a FETI--DP-type method is proposed and analyzed. It is established that the condition number of the preconditioned linear system is estimated by C(1 + maxi logHi/hi)² with a constant C independent of hi, hi/hj, Hi and the jumps of coefficients. The method is well suited for parallel computations and it can be extended to three-dimensional problems. Numerical results are presented to validate the theory. [ABSTRACT FROM AUTHOR]

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]