Discontinuous Galerkin derivative operators with applications to second‐order elliptic problems and stability

A discontinuous Galerkin (DG) finite‐element interior calculus is used as a common framework to describe various DG approximation methods for second‐order elliptic problems. Using the framework, symmetric interior‐penalty methods, local discontinuous Galerkin methods, and dual‐wind discontinuous Galerkin methods will be compared by expressing all of the methods in primal form. The penalty‐free nature of the dual‐wind discontinuous Galerkin method will be both motivated and used to better understand the analytic properties of the various DG methods. Consideration will be given to Neumann boundary conditions with numerical experiments that support the theoretical results. Many norm equivalencies will be derived laying the foundation for applying dual‐winding techniques to other problems. Copyright © 2015 John Wiley & Sons, Ltd.     

基于正交多项式下的数值微分任意阶稳定逼近

本文研究了数值微分问题.利用基于正交多项式理论下的积分算子方法,获得了可以稳定逼近已知函数任意阶导数的结果,推广了Lanczos积分方法的结果.     

Sensitivity of shear rate in artificial grafts using automatic differentiation

An accurate numerical simulation of blood requires the solution of incompressible Navier–Stokes equations coupled with specific constitutive models. We consider a generalized Newtonian fluid model in which viscosity depends on shear rate, accounting for the shear‐thinning behavior of blood. Previous work on the design of an artificial graft indicated that there is an influence of the fluid model on the solution of the partial differential equation‐constrained shape optimization problem. Therefore, we carry out a sensitivity analysis of the actual implementation of the flow solver using automatic differentiation (AD). We compare the sensitivities of shear rate with respect to viscosity for different configurations and validate the truncation‐error‐free sensitivities obtained from AD with those based on divided differencing and, if available, with analytic derivatives. Copyright © 2009 John Wiley & Sons, Ltd.     

Effect of random perturbations on adaptive observation techniques

An observation sensitivity (OS) method to identify targeted observations is implemented in the context of four‐dimensional variational (4D‐Var) data assimilation. This methodology is compared with the well‐established adjoint sensitivity (AS) method using a nonlinear Burgers equation as a test model. Automatic differentiation software is used to implement the first‐order adjoint model (ADM) to calculate the gradient of the cost function required in the 4D‐Var minimization algorithm and in the AS computations and the second‐order ADM to obtain information on the Hessian matrix of the 4D‐Var cost that is necessary in the OS computations. Numerical results indicate that the observation‐targeting is particularly successful in reducing the forecast error for moderate Reynolds numbers. The potential benefits of the OS targeting approach over the AS are investigated. The effect of random perturbations on the performance of these adaptive observation techniques is also analyzed. Copyright © 2011 John Wiley & Sons, Ltd.     

Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

The initial-value problem for a first-order evolution equation is discretised in time by means of the two-step backward differentiation formula (BDF) on a variable time grid. The evolution equation is governed by a monotone and coercive potential operator. On a suitable sequence of time grids, the piecewise constant interpolation and a piecewise linear prolongation of the time discrete solution are shown to converge towards the weak solution if the ratios of adjacent step sizes are close to 1 and do not vary too much.      

Effects of Bisphenol A and Its Alternatives,Bisphenol F and Tetramethyl Bisphenol F on Osteoclast Differentiation

Bisphenol A (BPA) is a typical environmental endocrine disruptor that exhibits estrogen-mimicking, hormone-like properties and can cause the collapse of bone homeostasis by an imbalance between osteoblasts and osteoclasts. Various BPA substitutes, structurally similar to BPA, have been used to manufacture ‘BPA-free’ products; however, the regulatory role of BPA alternatives in osteoclast differentiation still remains unelucidated. This study aimed to investigate the effects of these chemicals on osteoclast differentiation using the mouse osteoclast precursor cell line RAW 264.7. Results confirmed that both BPA and its alternatives, bisphenol F and tetramethyl bisphenol F (TMBPF), were nontoxic to RAW 264.7 cells. In particular, tartrate-resistant acid phosphatase (TRAP)-positive multinucleated cell staining and activity calculation assays revealed that TMBPF enhanced osteoclast differentiation upon stimulation of the receptor activator of nuclear factor-kappa B ligand (RANKL). Additionally, TMBPF activated the mRNA expression of osteoclast-related target genes, such as the nuclear factor of activated T-cells, cytoplasmic 1 (NFATc1), tartrate-resistant acid phosphatase (TRAP), and cathepsin K (CtsK). Western blotting analysis indicated activation of the mitogen-activated protein kinase signaling pathway, including phosphorylation of c-Jun N-terminal kinase and p38. Together, the results suggest that TMBPF enhances osteoclast differentiation, and it is critical for bone homeostasis and skeletal health.