Divergence Operator Research Articles

Mimetic finite difference operators and higher order quadratures

Mimetic finite difference operators textbf{D}, textbf{G} are discrete analogs of the continuous divergence (div) and gradient (grad) operators. In the discrete sense, these discrete operators satisfy the same properties as those of their continuum counterparts. In particular, they satisfy a discrete extended Gauss’ divergence theorem. This paper investigates the higher-order quadratures associated with the fourth- and sixth- order mimetic finite difference operators, and show that they are indeed numerical quadratures and satisfy the divergence theorem. In addition, extensions to curvilinear coordinates are treated. Examples in one and two dimensions to illustrate numerical results are presented that confirm the validity of the theoretical findings.

Strong uniqueness of finite-dimensional Dirichlet operators with singular drifts

We show [Formula: see text]-uniqueness for any [Formula: see text] and the essential self-adjointness of a Dirichlet operator [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text]. In particular, [Formula: see text] is allowed to be in [Formula: see text] or in [Formula: see text] for some [Formula: see text], while [Formula: see text] is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.

An improved elastic wavefield separation method based on the Helmholtz decomposition

Elastic wave-mode separation plays an important role in elastic reverse time migration and elastic full-waveform inversion. It helps to remove crosstalk artifacts and improve imaging quality. A class of efficient methods for elastic wave-mode separation in isotropic elastic media is the Helmholtz decomposition technique. Although this kind of approach produces pure-mode vector wavefields with correct amplitudes, phases, and physical units, their computational costs are still high especially for 3D large-scale problems. By making use of the relationships among the divergence, curl, gradient, and exterior derivative operations, we develop an improved elastic wave-mode separation based on the Helmholtz decomposition. We also need to solve a Poisson equation, but the Laplace operator operates on a scalar function rather than a vector function. Thus, for multidimensional (2D or 3D) problems, the Poisson equation only needs to be solved once for the vector P and S wavefields. This allows us to reduce the computational cost of the conventional Helmholtz decomposition method by a factor of two for solving 2D problems. For a 3D problem, the computational cost can be reduced by a factor of three. To further reduce the computational cost, by introducing a smooth extension technique, we transform the problem into the wavenumber domain via the Fourier transform and use a fast solver for the Poisson equation. The resulting wavefield separation method not only produces P and S waves with the same phases and amplitudes as the input-coupled wavefields but also significantly reduces the computational cost. Numerical tests indicate the efficiency of the proposed method and confirm the theoretical results.

An L2-finite element approximation of the solution to div/curl-systems with nonempty null space

Partial differential equations involving divergence or curl operators often introduce nonempty null-space components in their solutions, causing conventional numerical methods to produce inaccurate results. In this paper, we present a new algorithm that addresses this challenge by explicitly accounting for the null-space component. The dual system least squares method is used to identify this component and remove it from the approximation using negative-norm minimization. Our approach retains the use of H1-conforming basis functions, ensuring that the resulting approximation is orthogonal to the null space of the operator. The effectiveness of our algorithm is demonstrated through numerical experiments with various partial differential equations, showing improved accuracy compared to conventional methods.

Nodal discrete duality numerical scheme for nonlinear diffusion problems on general meshes

Abstract Discrete duality finite volume (DDFV) schemes are known for their ability to approximate nonlinear and linear anisotropic diffusion operators on general meshes, but they possess several drawbacks. The most important drawback of DDFV is the simultaneous use of the cell and the node unknowns. We propose a discretization approach that incorporates DDFV ideas and the associated analysis techniques, but allows for a rapid elimination of the cell unknowns. Further, unlike the DDFV scheme, the new ‘Nodal Discrete Duality’ (NDD) scheme does not require specific adaptation in presence of discontinuities of the diffusion tensor along cell boundaries. We describe in detail the 2D NDD framework and its two 3D variants, focusing on the consistency properties of the discrete gradient and discrete divergence operators and on the key structural property of discrete duality. For the 2D scheme, convergence analysis is carried out and a series of numerical tests are provided for a large family of nonlinear anisotropic elliptic problems with zero Dirichlet boundary condition.

The Effects of Enfacement Illusion on Self and Other Perception and Relevant Implications

Perceiving self is one of the crucial issues discussed in psychology. Studies have suggested that the evaluation of the self could be manipulated by divergent factors and operations. Some experiments proved the variability of self-concept. A typical instance is the rubber hand illusion, causing a discrepancy between the visual and tactile sensation and the confusion of identity. Enfacement illusion, similarly, have the resembling influence but focuses on the human face instead of the hand. Its an important phenomenon with the significant meaning of being related to the perception of self and other. The impact of the enfacement effect on one's perspective of oneself and that of others was examined through the analysis of recent research, with further implications given. Discussions done in this paper could make contributions in the area of comprehension of self and other. First, various fingdings bring up the individualized irregularity that emerged among people. Second, according to plenty of studies, enfacement has the potential to encourage the establishment of relationships with more intimacy. Thirdly, studies on illusions other than the perception of faces were conducted, and they demonstrated similarities to the enfacement illusion's effect. Limitations of not elaborating on compound factors associated exist in previous research. Addition explorations are required for examining the causation of values altered. This review can provide some suggestions for intervention studies and applications in children with social functioning deficits.

Binary thermal fluids computation over arbitrary surfaces with second-order accuracy and unconditional energy stability based on phase-field model

In this paper, we propose an efficient surface computational system with second-order spatial and temporal accuracy to solve multiple physical field coupling problems over arbitrary surfaces. The computational system is coupled with heat transfer equation and incompressible Navier–Stokes equation based on a phase-field model. Due to the discretization of the triangular grids, we define the discretized gradient operator, divergence operator and the Laplace–Beltrami operator with second-order spatial accuracy. The Crank–Nicolson-type scheme is used to confirm the temporal accuracy. The Navier–Stokes equation is solved by the projection method. We use the biconjugate gradient stabilized method to solve the involved equations. The discrete system is provable to be unconditionally stable and the mass conservation law is satisfied during the computation, which implies that the proposed method is not limited by the temporal step. Several computational tests are conducted to show the efficiency, robustness and accuracy of the proposed scheme.

Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: Discrete divergence operator

A least-squares neural network (LSNN) method was introduced for solving scalar linear and nonlinear hyperbolic conservation laws (HCLs) in Cai et al. (2021, 2022). This method is based on an equivalent least-squares (LS) formulation and uses ReLU neural network as approximating functions, making it ideal for approximating discontinuous functions with unknown interface location. In the design of the LSNN method for HCLs, the numerical approximation of differential operators is a critical factor, and standard numerical or automatic differentiation along coordinate directions can often lead to a failed NN-based method. To overcome this challenge, this paper rewrites HCLs in their divergence form of space and time and introduces a new discrete divergence operator. As a result, the proposed LSNN method is free of penalization of artificial viscosity.Theoretically, the accuracy of the discrete divergence operator is estimated even for discontinuous solutions. Numerically, the LSNN method with the new discrete divergence operator was tested for several benchmark problems with both convex and non-convex fluxes, and was able to compute the correct physical solution for problems with rarefaction, shock or compound waves. The method is capable of capturing the shock of the underlying problem without oscillation or smearing, even without any penalization of the entropy condition, total variation, and/or artificial viscosity.

Weak Galerkin finite element method for linear poroelasticity problems

In this paper, we develop a weak Galerkin (WG) finite element method for a linear poroelasticity model where weak divergence and weak gradient operators defined over discontinuous functions are introduced. We establish both the continuous and discrete time WG schemes, and obtain their optimal convergence order estimates in a discrete H1 norm for the displacement and in H1 and L2 norms for the pressure. Finally, we present some numerical experiments on different kinds of meshes to illustrate the theoretical error estimates, and furthermore verify the locking-free property of our proposed method.

A Survey on Discrete Laplacians for General Polygonal Meshes

AbstractThe Laplace Beltrami operator is one of the essential tools in geometric processing. It allows us to solve numerous partial differential equations on discrete surface meshes, which is a fundamental building block in many computer graphics applications. Discrete Laplacians are typically limited to standard elements like triangles or quadrilaterals, which severely constrains the tessellation of the mesh. But in recent years, several approaches were able to generalize the Laplace Beltrami and its closely related gradient and divergence operators to more general meshes. This allows artists and engineers to work with a wider range of elements which are sometimes required and beneficial in their field. This paper discusses the different constructions of these three ubiquitous differential operators on arbitrary polygons and analyzes their individual advantages and properties in common computer graphics applications.

Analysis of divergence free conforming virtual elements for the Brinkman problem

In this paper, we develop stability analysis, including inverse inequality, [Formula: see text] norm equivalence and interpolation error estimates, for divergence free conforming virtual elements in arbitrary dimension. A local energy projector based on the local Stokes problem is suggested, which commutes with the divergence operator. After defining a discrete bilinear form and a stabilization involving only the boundary degrees of freedom (DoF) and parts of the interior DoF, a new divergence free conforming virtual element method is advanced for the Brinkman problem, which can be reduced to a simpler method due to the divergence free discrete velocity. An optimal convergence rate is derived for the discrete method. Furthermore, we achieve a uniform half convergence rate of the discrete method in consideration of the boundary layer phenomenon. Finally, some numerical results are provided to validate the convergence of the discrete method.

Locally structure-preserving div-curl operators for high order discontinuous Galerkin schemes

We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG) operator that recovers at the discrete level the algebraic property related to the divergence of the curl of a vector field, which is typically referred to as div-curl problem. A staggered Cartesian grid is adopted in 3D, where the vector field is naturally defined at the corners of the control volume, while its curl is evaluated as a cell-centered quantity. Firstly, the curl operator is rewritten as the divergence of a tensor, hence allowing compatible finite difference schemes to be devised and to be proven to mimic the algebraic div-curl property. Successively, a high order DG divergence operator is built upon integration by parts, so that the structure-preserving finite difference div-curl operator is exactly retrieved for first order discretizations. We further demonstrate that the novel SPDG schemes are capable of obtaining a zero div-curl identity with machine precision from second up to sixth order accuracy. Curl-grad operators follow as a consequence. In a second part, we show the applicability of these SPDG methods by solving the incompressible Navier-Stokes equations written in vortex-stream formulation. This hyperbolic system deals with divergence-free involutions related to the velocity and vorticity field as well as to the stream function, thus it provides an ideal setting for the validation of the novel schemes. A compatible discretization of the numerical viscosity is also proposed in order to maintain the structure-preserving property of the div-curl DG operators even in the presence of artificial or physical dissipative terms. Finally, to overcome the time step restriction dictated by the viscous sub-system, Implicit-Explicit (IMEX) Runge-Kutta time stepping techniques are tailored to handle the SPDG framework. Numerical examples show that the theoretical order of convergence is reached by this new class of methods, and prove the exact preservation of the incompressibility constraint.

Construction of a Right Inverse for the Divergence in Non-cylindrical Time Dependent Domains

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Hölder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Hölder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. The most precise estimates are given in weighted spaces, where the weight depends on the distance to the boundary. This allows for the deficit to be captured precisely in the vicinity of irregularities of the boundary. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier–Stokes equations in time dependent domains.

Parallel Overlapping Community Detection Algorithm on GPU

Community detection is one of the most representative graph mining applications, which is often assembled as a concurrent graph partition application to explore the maximum modularity (or gained modularity) of each community. However, many branch divergence operations create significant obstacles to unleashing GPU&#x2019;s high throughput and memory bandwidth, which are needed in community detection applications to divide the vertices into different communities. In this paper, we present Lugger, a GPU-based overlapping community detection algorithm that reduces GPU&#x2019;s branch divergence via the customer-designed cache-aware parallel searching technique. In Lugger, we first design a cache-aware parallel searching policy using the B-Tree structure. Then, we set the B-Tree node matches with the GPU cache line to meet the coalesced memory access manner and avoid the branch divergence in warps. Moreover, we design a positive node splitting scheme to reduce the lock operation and idle threads when building the B-Tree structure. In addition, we implement a warp-centric thread assignment strategy to make sure the workloads across threads are balanced. We implement the proposed algorithm on NVIDIA GPU and evaluate the performance on eight large graphs (up to <inline-formula><tex-math notation="LaTeX">$3M$</tex-math></inline-formula> vertices and <inline-formula><tex-math notation="LaTeX">$117M$</tex-math></inline-formula> edges) with ground-truth communities. The experimental results show that Lugger can outperform the state-of-the-art works on scalability and detection quality.

Managing start-up – incumbent digital solution co-creation: a four-phase process for intermediation in innovative contexts

ABSTRACT As incumbents strive to collaborate with start-ups in the pursuit of cutting-edge digital solutions, the complexities posed by disparate partners and their innovative endeavours often lead to intricate tensions. Our research underscores the critical role of innovation intermediaries in enabling a successful digital co-creation, yet a deeper understanding of this novel and evolving context is required. Through a comprehensive study of two innovation intermediaries, five incumbent companies, and eleven start-ups, we shed light on how intermediaries can effectively mitigate the hard-to-manage tensions that emerge. Our analysis uncovers three primary tensions: incompatible digital co-creation cultures, divergent digital innovation operations, and misaligned technical capabilities. We further propose a four-phase process for innovation intermediation, including the establishment of digital co-creation foundations, catalysing digital innovation projects, orchestrating the co-creation process, and scaling the resulting outcomes.

A nodal integration based two level local projection meshfree stabilization method for convection diffusion problems

This paper presents a new two level local projection meshfree stabilization (LPMS) method under the classical stabilized conforming nodal integration (SCNI) framework to solve convection diffusion problems suitable to a linear order approximation. Dropping the higher order terms in the residual based Petrov-Galerkin stabilization methods, usual in the Gauss integration, produces erroneous results if the classical SCNI is used because of the product of a quantity and its smoothed gradient (SCNI divergence operator) in the convective term. LPMS method is an alternative way to achieve a stable solution with the linear order approximation. In the present method, the design of the stabilization of convective terms stems from the fine scales defined on the subcells of Voronoi cells of SCNI. This LPMS with SCNI combination excludes the computational need for derivatives. First order mesh basis functions are sufficient as the SCNI replaces lower order derivatives with smoothed gradients, while local projection stabilization (LPS) excludes the terms with higher order derivatives. The present LMPS method is tested against standard benchmark problems with different distorted grids and compared with three existing stabilization methods. It is found to be in good agreement with the classical numerical methods for problems with thin layers.

Partial Euler operators and the efficient inversion of Div

AbstractThe problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.

On Scaling Properties for Two-State Problems and for a Singularly Perturbed T_{3} Structure

In this article we study quantitative rigidity properties for the compatible and incompatible two-state problems for suitable classes of mathcal{A}-free differential inclusions and for a singularly perturbed T_{3} structure for the divergence operator. In particular, in the compatible setting of the two-state problem we prove that all homogeneous, first order, linear operators with affine boundary data which enforce oscillations yield the typical epsilon ^{frac{2}{3}}-lower scaling bounds. As observed in Chan and Conti (Math. Models Methods Appl. Sci. 25(06):1091–1124, 2015) for higher order operators this may no longer be the case. Revisiting the example from Chan and Conti (Math. Models Methods Appl. Sci. 25(06):1091–1124, 2015), we show that this is reflected in the structure of the associated symbols and that this can be exploited for a new Fourier based proof of the lower scaling bound. Moreover, building on Rüland and Tribuzio (Arch. Ration. Mech. Anal. 243(1):401–431, 2022); Garroni and Nesi (Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460(2046):1789–1806, 2004, https://doi.org/10.1098/rspa.2003.1249); Palombaro and Ponsiglione (Asymptot. Anal. 40(1):37–49, 2004), we discuss the scaling behavior of a T_{3} structure for the divergence operator. We prove that as in Rüland and Tribuzio (Arch. Ration. Mech. Anal. 243(1):401–431, 2022) this yields a non-algebraic scaling law.

Divergent Protocol for the Synthesis of Isoquinolino[1,2-b]quinazolinone and Isoquinolino[2,1-a]quinazolinone Derivatives.

The development of robust and step-economic strategies to access structurally diverse drug-like compound collections remains a challenge. A distinct structural option that constitutes the core scaffold of many biologically significant molecules is the quinazolinone ring system. Several members of this family of privileged substructures have gained attention due to their diverse biological activities. In this context, the development of an efficient strategy for their access is needed. Herein, we report a divergent metal-free operation to access a diverse collection of C6-substituted pyrrolo[4',3',2':4,5]isoquinolino[1,2-b]quinazolin-8(6H)-one and pyrrolo[4',3',2':4,5]isoquinolino[2,1-a]quinazolin-12(6H)-one architectures. The described cascade unites Friedel-Crafts and aza-Michael addition reactions. This operationally simple protocol enables a rapid access to these scaffolds and is compatible with a wide scope of starting materials. In addition, the cascade features a promising approach for the design of unique compound libraries for drug design and discovery programs.

Mimetic finite differences for boundaries misaligned with grid nodes

This paper considers the problem of mimetic staggered-grid finite difference scheme construction in the case of a boundary misaligned with the grid points. We show that depending on the reciprocal positions of the nearest grid point and whether the point is inside or outside the domain, two cases should be considered separately. In both cases, we present general rules to construct the mimetic difference operators to approximate the gradient and divergence operators. In addition, we investigate the stability criterion of the explicit in time scheme, which employs derived operators to approximate spatial derivatives. We show that, depending on the reciprocal positions of the nearest grid point, the Courant number may be smaller by 7% than that of the initial value problem approximation.