Fortin operator for the Taylor–Hood element

We design a local Fortin operator for the lowest-order Taylor–Hood element in any dimension, which was previously constructed only in 2D. In the construction we use tangential edge bubble functions for the divergence correcting operator. This naturally leads to an alternative inf-sup stable reduced finite element pair. Furthermore, we provide a counterexample to the inf-sup stability and hence to existence of a Fortin operator for the $$P_2$$ P 2 –$$P_0$$ P 0 and the augmented Taylor–Hood element in 3D.

A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.

Visual Servo Control System of a Piezoelectric2-Degree-of-Freedom Nano-Stepping Motor

A nano-stepping motor can translate or rotate when its piezoelectric element pair is electrically driven in-phase or anti-phase. It offers millimeter-level stroke, sub-micron-level stepping size, and sub-nanometer-level scanning resolution. This article proposes a visual servo system to control the nano-stepping motor, since its stepping size is not consistent due to changing contact friction, using a custom built microscopic instrument and image recognition software. Three kinds of trajectories—straight lines, circles, and pentagrams—are performed successfully. The smallest straightness and roundness ever tested are 0.291 µm and 2.380 µm. Experimental results show that the proposed controller can effectively compensate for the error and precisely navigate the rotor along a desired trajectory.

Parallel two-step finite element algorithm based on fully overlapping domain decomposition for the time-dependent natural convection problem

Purpose This paper aims to propose two parallel two-step finite element algorithms based on fully overlapping domain decomposition for solving the 2D/3D time-dependent natural convection problem. Design/methodology/approach The first-order implicit Euler formula and second-order Crank–Nicolson formula are used to time discretization respectively. Each processor of the algorithms computes a stabilized solution in its own global composite mesh in parallel. These algorithms compute a nonlinear system for the velocity, pressure and temperature based on a lower-order element pair (P1b-P1-P1) and solve a linear approximation based on a higher-order element pair (P2-P1-P2) on the same mesh, which shows that the new algorithms have the same convergence rate as the two-step finite element methods. What is more, the stability analysis of the proposed algorithms is derived. Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms. Findings Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms. Originality/value The novel parallel two-step algorithms for incompressible natural convection problem are proposed. The rigorous analysis of the stability is given for the proposed parallel two-step algorithms. Extensive 2D/3D numerical tests demonstrate that the parallel two-step algorithms can deal with the incompressible natural convection problem for high Rayleigh number well.

Parallel two-step finite element algorithm for the stationary incompressible magnetohydrodynamic equations

Purpose The purpose of this paper is to propose a local parallel finite element algorithm based on fully overlapping domain decomposition technique to solve the incompressible magnetohydrodynamic equations. Design/methodology/approach The algorithm uses a lower-order element pair to compute an initial approximation by the Oseen-type iteration and uses a higher-order element pair to solve a linear system in each processor. Findings Besides, the convergence analysis of local parallel finite element algorithm is given. Finally, numerical experiments are presented to verify the efficiency of the proposed algorithm. Originality/value Compared with the numerical solution of the common two-step method, this method is easy to realize and can produce a more accurate solution. And, this approach is executed in parallel, so it saves a lot of computational time.

Synergistic binding of bHLH transcription factors to the promoter of the maize NADP-ME gene used in C4 photosynthesis is based on an ancient code found in the ancestral C3 state

C4 photosynthesis has evolved repeatedly from the ancestral C3 state to generate a carbon concentrating mechanism that increases photosynthetic efficiency. This specialised form of photosynthesis is particularly common in the PACMAD clade of grasses, and is used by many of the world’s most productive crops. The C4 cycle is accomplished through cell-type specific accumulation of enzymes but cis-elements and transcription factors controlling C4 photosynthesis remain largely unknown. Using the NADP-Malic Enzyme (NADP-ME) gene as a model we aimed to better understand molecular mechanisms associated with the evolution of C4 photosynthesis. Two basic Helix-Loop-Helix (bHLH) transcription factors, ZmbHLH128 and ZmbHLH129, were shown to bind the C4NADP-ME promoter from maize. These proteins form heterodimers and ZmbHLH129 impairs trans-activation by ZmbHLH128. Electrophoretic mobility shift assays indicate that a pair of cis-elements separated by a seven base pair spacer synergistically bind either ZmbHLH128 or ZmbHLH129. This pair of cis-elements is found in both C3 and C4 species of the PACMAD clade. Our analysis is consistent with this cis-element pair originating from a single motif present in the ancestral C3 state. We conclude that C4 photosynthesis has co-opted an ancient C3 regulatory code built on G-box recognition by bHLH to regulate the NADP-ME gene. More broadly, our findings also contribute to the understanding of gene regulatory networks controlling C4 photosynthesis.

Stabilised finite element methods for the Oseen problem on anisotropic quadrilateral meshes

In this work we present and analyse new inf-sup stable, and stabilised, finite element methods for the Oseen equation in anisotropic quadrilateral meshes. The meshes are formed of closed parallelograms, and the analysis is restricted to two space dimensions. Starting with the lowest order ℚ12 × ℙ0 pair, we first identify the pressure components that make this finite element pair to be non-inf-sup stable, especially with respect to the aspect ratio. We then propose a way to penalise them, both strongly, by directly removing them from the space, and weakly, by adding a stabilisation term based on jumps of the pressure across selected edges. Concerning the velocity stabilisation, we propose an enhanced grad-div term. Stability and optimal a priori error estimates are given, and the results are confirmed numerically.

An Efficient Algorithm with Stabilized Finite Element Method for the Stokes Eigenvalue Problem

This paper provides a two-space stabilized mixed finite element scheme for the Stokes eigenvalue problem based on local Gauss integration. The two-space strategy contains solving one Stokes eigenvalue problem using theP1-P1finite element pair and then solving an additional Stokes problem using theP2-P2finite element pair. The postprocessing technique which increases the order of mixed finite element space by using the same mesh can accelerate the convergence rate of the eigenpair approximations. Moreover, our method can save a large amount of computational time and the corresponding convergence analysis is given. Finally, numerical results are presented to confirm the theoretical analysis.