No Arabic abstract
By supplementing the pressure space for the Taylor-Hood element a triangular element that satisfies continuity over each element is produced. Making a novel extension of the patch argument to prove stability, this element is shown to be globally stable and give optimal rates of convergence on a wide range of triangular grids. This theoretical result is extended in the discussion given in the appendix, showing how optimal convergence rates can be obtained on all grids. Two examples are presented, one illustrating the convergence rates and the other illustrating difficulties with the Taylor-Hood element which are overcome by the element presented here.
This paper will suggest a new finite element method to find a $P^4$-velocity and a $P^3$-pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a $P^4$-velocity. Then, using the calculated velocity, a locally calculable $P^3$-pressure will be defined component-wisely. The resulting $P^3$-pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the $P^4$-velocity. Besides, the method overcomes the problem of singular vertices or corners.
The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334] is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the $L^2$-norm and in a modified energy norm, as well as a reduced convergence order of ${cal O}(h^{3/2})$ in the standard $H^1$-norm. Finally, we present numerical examples to substantiate the theoretical findings.
We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babuvska-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.
A finite element elasticity complex on tetrahedral meshes is devised. The $H^1$ conforming finite element is the smooth finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an $H(textrm{inc})$-conforming finite element for symmetric tensors is the main focus of this paper. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the $textrm{inc}$ operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition of polynomial tensor spaces. The trace of the $textrm{inc}$ operator is induced from a Greens identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Our construction appears to be the first $H(textrm{inc})$-conforming finite elements on tetrahedral meshes without further splits.
We propose a new discretization of a mixed stress formulation of the Stokes equations. The velocity $u$ is approximated with $H(operatorname{div})$-conforming finite elements providing exact mass conservation. While many standard methods use $H^1$-conforming spaces for the discrete velocity, $H(operatorname{div})$-conformity fits the considered variational formulation in this work. A new stress-like variable $sigma$ equalling the gradient of the velocity is set within a new function space $H(operatorname{curl} operatorname{div})$. New matrix-valued finite elements having continuous normal-tangential components are constructed to approximate functions in $H(operatorname{curl} operatorname{div})$. An error analysis concludes with optimal rates of convergence for errors in $u$ (measured in a discrete $H^1$-norm), errors in $sigma$ (measured in $L^2$) and the pressure $p$ (also measured in $L^2$). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.