Finite Element Iterative Methods for the 3D Steady Navier–Stokes Equations

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair Xh×Mh. The method consists of transmitting the finite element solution (uh,ph) of the 3D steady Navier–Stokes equations into the finite element solution pairs (uhn,phn) based on the finite element space pair Xh×Mh of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair Xh×Mh satisfies the discrete inf-sup condition in a 3D domain Ω. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution (uhn,phn) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to (σ,h) of the FE solution (uhn,phn) to the exact solution (u,p) of the 3D steady Navier–Stokes equations in the H1−L2 norm. Finally, we also give the convergence order with respect to (σ,h) of the FE velocity uhn to the exact velocity u of the 3D steady Navier–Stokes equations in the L2 norm.

A family of C1 quadrilateral finite elements

We present a novel family of C1 quadrilateral finite elements, which define global C1 spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by Brenner and Sung (J. Sci. Comput. 22(1-3), 83-118, 2005), which is based on polynomial elements of tensor-product degree p ≥ 6, to all degrees p ≥ 3. The proposed C1 quadrilateral is based upon the construction of multi-patch C1 isogeometric spaces developed in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles, developed in Argyris et al. (Aeronaut. J. 72(692), 701–709 1968). Just as for the Argyris triangle, we additionally impose C2 continuity at the vertices. In contrast to Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019), in this paper, we concentrate on quadrilateral finite elements, which significantly simplifies the construction. We present macro-element constructions, extending the elements in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005), for polynomial degrees p = 3 and p = 4 by employing a splitting into 3 × 3 or 2 × 2 polynomial pieces, respectively. We moreover provide approximation error bounds in $L^{\infty }$ L ∞ , L2, H1 and H2 for the piecewise-polynomial macro-element constructions of degree p ∈{3,4} and polynomial elements of degree p ≥ 5. Since the elements locally reproduce polynomials of total degree p, the approximation orders are optimal with respect to the mesh size. Note that the proposed construction combines the possibility for spline refinement (equivalent to a regular splitting of quadrilateral finite elements) as in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 30) with the purely local description of the finite element space and basis as in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005). In addition, we describe the construction of a simple, local basis and give for p ∈{3,4,5} explicit formulas for the Bézier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed C1 quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for p = 5) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.

From Continent to Ocean: Investigating the Multi-Element and Precious Metal Geochemistry of the Paraná-Etendeka Large Igneous Province Using Machine Learning Tools

Large Igneous Provinces, and by extension the mantle plumes that generate them, are frequently associated with platinum-group element (PGE) ore deposits, yet the processes controlling the metal budget in plume-derived magmas remains debated. In this paper, we present a new whole-rock geochemical data set from the 135 Ma Paraná-Etendeka Large Igneous Province (PELIP) in the South Atlantic, which includes major and trace elements, PGE, and Au concentrations for onshore and offshore lavas from different developmental stages in the province, which underwent significant syn-magmatic continental rifting from 134 Ma onwards. The PELIP presents an opportunity to observe magma geochemistry as the continent and sub-continental lithospheric mantle (SCLM) are progressively removed from a melting environment. Here, we use an unsupervised machine learning approach (featuring the PCA, t-SNE and k-means clustering algorithms) to investigate the geochemistry of a set of (primarily basaltic) onshore and offshore PELIP lavas. We test the hypothesis that plume-derived magmas can scavenge precious metals including PGE from the SCLM and explore how metal concentrations might change the metal content in intraplate magmas throughout rifting. Onshore lavas on the Etendeka side of the PELIP are classified as the products of deep partial melts of the mantle below the African craton but without significant PGE enrichment. Offshore lavas on both continents exhibit similarities through the multi-element space to their onshore equivalents, but they again lack PGE enrichment. Of the four onshore lava types on the Paraná side of the PELIP, the Type 1 (Southern) and Type 1 (Central-Northern) localities exhibit separate PGE-enriched assemblages (Ir-Ru-Rh and Pd-Au-Cu, respectively). It follows that there is a significant asymmetry to the metallogenic character of the PELIP, with enrichment focused specifically on lavas from the South American continent edge in Paraná. This asymmetry contrasts with the North Atlantic Igneous Province (NAIP), a similar geodynamic environment in which continent-edge lavas are also PGE-enriched, albeit on both sides of the plume-rift system. We conclude that, given the similarities in PGE studies of plume-rift environments, SCLM incorporation under progressively shallowing (i.e., rifting) asthenospheric conditions promotes the acquisition of metasomatic and residual PGE-bearing minerals, boosting the magma metal budget.

Low complexity sparse beamspace DOA estimation via single measurement vectors for uniform circular array

In this paper, we present a low complexity sparse beamspace direction-of-arrival (DOA) estimation method for uniform circular array (UCA). In the proposed method, we firstly use the beamspace transformation (BT) to transform the signal model of UCA in element-space domain to that of virtual uniform linear array (ULA) in beamspace domain. Subsequently, by applying the vectoring operator on the virtual ULA-like array signal model, a novel dimension-reduction sparse beamspace signal model is derived based on Khatri-Rao (KR) product, the observation data of which is represented by the single measurement vectors (SMVs) via vectorization of sparse covariance matrix. And then, the DOA estimation is formulated as a convex optimization problem by following the concept of a sparse-signal-representation (SSR) of the SMVs. Finally, simulations are carried out to validate the effectiveness of the proposed method. The results show that without knowledge of the number of signals, the proposed method not only has higher DOA resolution than the subspace-based methods in low signal-to-noise ratio (SNR), but also has far lower computational complexity than other sparse-like DOA estimation methods.

Vertex Displacement-Based Discontinuous Deformation Analysis Using Virtual Element Method

To avoid disadvantages caused by rotational degrees of freedom in the original Discontinuous Deformation Analysis (DDA), a new block displacement mode is defined within a time step, where displacements of all the block vertices are taken as the degrees of freedom. An individual virtual element space V1(Ω) is defined for a block to illustrate displacement of the block using the Virtual Element Method (VEM). Based on VEM theory, the total potential energy of the block system in DDA is formulated and minimized to obtain the global equilibrium equations. At the end of a time step, the vertex coordinates are updated by adding their incremental displacement to their previous coordinates. In the new method, no explicit expression for the displacement u is required, and all numerical integrations can be easily computed. Four numerical examples originally designed by Shi are analyzed, verifying the effectiveness and precision of the proposed method.

Reliability and Efficiency of DWR-Type A Posteriori Error Estimates with Smart Sensitivity Weight Recovering

We derive efficient and reliable goal-oriented error estimations, and devise adaptive mesh procedures for the finite element method that are based on the localization of a posteriori estimates. In our previous work [B. Endtmayer, U. Langer and T. Wick, Two-side a posteriori error estimates for the dual-weighted residual method, SIAM J. Sci. Comput. 42 2020, 1, A371–A394], we showed efficiency and reliability for error estimators based on enriched finite element spaces. However, the solution of problems on an enriched finite element space is expensive. In the literature, it is well known that one can use some higher-order interpolation to overcome this bottleneck. Using a saturation assumption, we extend the proofs of efficiency and reliability to such higher-order interpolations. The results can be used to create a new family of algorithms, where one of them is tested on three numerical examples (Poisson problem, p-Laplace equation, Navier–Stokes benchmark), and is compared to our previous algorithm.

P1-nonconforming divergence-free finite element method on square meshes for Stokes equations

Recently, the P1-nonconforming finite element space over square meshes has been proved stable to solve Stokes equations with the piecewise constant space for velocity and pressure, respectively. In this paper, we will introduce its locally divergence-free subspace to solve the elliptic problem for the velocity only decoupled from the Stokes equation. The concerning system of linear equations is much smaller compared to the Stokes equations. Furthermore, it is split into two smaller ones. After solving the velocity first, the pressure in the Stokes problem can be obtained by an explicit method very rapidly.