Grad Div Research Articles

An efficient algebraic multi-resolution sampling approach to 3-D magnetotelluric modelling

SUMMARY Since electromagnetic (EM) fields diffuse more smoothly to greater depth, it physically makes sense to apply fine discretization to model structure at near surface with an increasingly coarser grid both in horizontal and vertical directions as the depth increases for the numerical solution of EM fields. For finite-difference magnetotelluric (MT) forward modelling on regular staggered grids, this can lead to difficulties in discretizing the curl–curl equation governing the EM diffusion in the earth at regions, where the grid resolution changes. In this paper, we propose an efficient algebraic multi-resolution sampling (MRS) method for MT forward modelling. In this method, we do not require the generation of physical subgrids and merely subsample the field components. The coefficient matrix for the subsampled components can be obtained by matrix multiplication based on the initial linear system of equations and field interpolation. To guarantee divergence-free current during the iterative solution process at low frequencies, which is considered crucial for the development of efficient iterative solvers, our forward modelling is based a regularization equation obtained by augmenting the curl–curl equation with an equivalent scaled grad–div operator for electrical fields (explicitly enforcing the divergence-free condition). The correctness of our algebraic MRS algorithm is examined based on a layered model. Its stability and efficiency is exploited by comparing our results with the forward modelling on unsampled staggered grids for the Dublin Test Model 1 (DTM1) and a complex model with realistic topography, indicating a time reduction of about 42–82 per cent.

A cutFEM divergence–free discretization for the stokes problem

We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott–Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O(h) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O(h−1), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates.

On the influence of the nonlinear term in the numerical approximation of Incompressible Flows by means of proper orthogonal decomposition methods

We consider proper orthogonal decomposition (POD) methods to approximate the incompressible Navier–Stokes equations. We study the case in which one discretization for the nonlinear term is used in the snapshots (that are computed with a full order method (FOM)) and a different discretization of the nonlinear term is applied in the POD method. We prove that an additional error term appears in this case, compared with the case in which the same discretization of the nonlinear term is applied for both the FOM and the POD methods. However, the added term has the same size as the error coming from the FOM so that the rate of convergence of the POD method is barely affected. We analyze the case in which we add grad–div stabilization to both the FOM and the POD methods because it allows to get error bounds with constants independent of inverse powers of the viscosity. We also study the case in which no stabilization is added. Some numerical experiments support the theoretical analysis.

Improved convergence of the Arrow–Hurwicz iteration for the Navier–Stokes equation via grad–div stabilization and Anderson acceleration

We consider two modifications of the Arrow–Hurwicz (AH) iteration for solving the incompressible steady Navier–Stokes equations for the purpose of accelerating the algorithm: grad–div stabilization, and Anderson acceleration. AH is a classical iteration for general saddle point linear systems and it was later extended to Navier–Stokes iterations in the 1970’s which has recently come under study again. We apply recently developed ideas for grad–div stabilization and divergence-free finite element methods along with Anderson acceleration of fixed point iterations to AH in order to improve its convergence. Analytical and numerical results show that each of these methods improves AH convergence, but the combination of them yields an efficient and effective method that is competitive with more commonly used solvers.

Error analysis of proper orthogonal decomposition data assimilation schemes with grad–div stabilization for the Navier–Stokes equations

The error analysis of a proper orthogonal decomposition (POD) data assimilation (DA) scheme for the Navier–Stokes equations is carried out. A grad–div stabilization term is added to the formulation of the POD method. Error bounds with constants independent on inverse powers of the viscosity parameter are derived for the POD algorithm. No upper bounds in the nudging parameter of the data assimilation method are required. Numerical experiments show that, for large values of the nudging parameter, the proposed method rapidly converges to the real solution, and greatly improves the overall accuracy of standard POD schemes up to low viscosities over predictive time intervals.

Construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes

Given a sequence of finite element spaces which form a de Rham sequence, we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The dual representations also need to satisfy the de Rham sequence on the domain boundary. The matrix which converts primal representations to dual representations – the Hodge matrix – is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even for high order methods. The differential operators for dual representations will be defined. Vector operations, grad, curl and div, for primal and dual representations are both topological and do not depend on the metric, i.e. the size and shape of the mesh or the order of the numerical method. Derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations. As illustration of the use of dual representations, the method will be applied to (i) a mixed formulation for the Poisson problem in 3D, (ii) it will be shown that this approach allows one to preserve the equivalence between Dirichlet and Neumann problems in the finite dimensional setting and, (iii) the method will be applied to the approximation of grad–div eigenvalue problem on affine and non-affine meshes.

Numerical analysis of CNLF modular Grad–Div stabilization method for time-dependent Navier–Stokes equations

This paper presents a modular Grad–Div(GD) algorithm based on Crank–Nicolson LeapFrog (CNLF) scheme for the non-stationary Navier–Stokes model, which is second-order accurate. The CNLF modular Grad–Div(CNLF-MGD) method has the obvious advantage that the computational efficiency will not crash or slow down with the large parameters. Stability and error estimate of the numerical method are given. Some numerical tests are proposed to justify the theoretical analysis.

Numerical analysis of two grad–div stabilization methods for the time-dependent Stokes/Darcy model

We aim to present two algorithms for the non-stationary Stokes/Darcy model. The first one is the standard grad–div stabilization scheme. The other one is a modular grad–div based on the standard Backward Euler code which does not crash or slow down for large value grad–div parameters. Both algorithms cannot only improve the efficiency and accuracy of calculation but also can improve mass conservation, while the modular algorithm can be better. We give a complete theoretical analysis of the stability and error estimations of the algorithms. Finally, the theoretical results are verified by numerical experiments and the advantages of adding grad–div stabilization terms are demonstrated.

Numerical comparisons of finite element stabilized methods for a 2D vortex dynamics simulation at high Reynolds number

In this paper, we consider up-to-date and classical Finite Element (FE) stabilized methods for time-dependent incompressible flows. All studied methods belong to the Variational MultiScale (VMS) framework. So, different realizations of stabilized FE-VMS methods are compared using a high Reynolds number vortex dynamics simulation. In particular, a fully Residual-Based (RB)-VMS method is compared with the classical Streamline-Upwind Petrov–Galerkin (SUPG) method together with grad–div stabilization, a standard one-level Local Projection Stabilization (LPS) method, and a recently proposed LPS method by interpolation. These procedures do not make use of the statistical theory of equilibrium turbulence, and no ad-hoc eddy viscosity modeling is required for all methods. Applications to the simulation of a high Reynolds number flow with vortical structures on relatively coarse grids are showcased, by focusing on a two-dimensional plane mixing-layer flow. Both Inf–Sup Stable (ISS) and Equal Order (EO) H1-conforming FE pairs are explored using a second-order semi-implicit Backward Differentiation Formula (BDF2) in time. Based on the numerical studies conducted, it is concluded that the SUPG method using EO–FE pairs performs best among all methods. Furthermore, there seems to be no reason to extend the SUPG method by the higher order terms of the RB-VMS method.

The analogue of grad–div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes

grad–div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad–div stabilization for Discontinuous Galerkin methods is studied. Here, the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Our contribution is twofold: first, we characterize the limit for arbitrarily large penalization parameters, which shows that the stabilized nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit; second, we extend these ideas to the case of non-simplicial meshes; here, broken grad–div stabilization must be used in addition to the normal velocity jump penalization, in order to get the desired pressure robustness effect. The analysis is performed for the Stokes equations, and more complex flows and Crouzeix–Raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings.

An efficient and modular grad–div stabilization

This paper presents two modular grad–div algorithms for calculating solutions to the Navier–Stokes equations (NSE). These algorithms add to an NSE code a minimally intrusive module that implements grad–div stabilization. The algorithms do not suffer from either solver breakdown or debilitating slow down for large values of grad–div parameters. Stability and optimal-order convergence of the methods are proven. Numerical tests confirm the theory and illustrate the benefits of these algorithms over a fully coupled grad–div stabilization.

A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier–Stokes equations

AbstractWe prove that for several inf-sup stable mixed finite elements, the solution of the Chorin/Temam projection methods for Navier–Stokes equations equipped with grad–div stabilization with parameter γ converge to the associated coupled method solution with rate γ

A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier–Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad–div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at Reτ=180 as well as 590.

Stability analysis of pressure correction schemes for the Navier–Stokes equations with traction boundary conditions

We present a stability analysis for two different rotational pressure correction schemes with open and traction boundary conditions. First, we provide a stability analysis for a rotational version of the grad–div stabilized scheme of Bonito et al. (2015). This scheme turns out to be unconditionally stable, provided the stabilization parameter is suitably chosen. We also establish a conditional stability result for the boundary correction scheme presented in Bänsch (2014). These results are shown by employing the equivalence between stabilized gauge Uzawa methods and rotational pressure correction schemes with traction boundary conditions.

Mixed finite element methods with convection stabilization for the large eddy simulation of incompressible turbulent flows

The variational multiscale method thought as an implicit large eddy simulation model for turbulent flows has been shown to be an alternative to the widely used physical-based models. This method is traditionally combined with equal-order velocity–pressure pairs, since it provides pressure stabilization. In this work, we consider a different approach, based on inf–sup stable elements and convection-only stabilization. In order to do so, we consider a symmetric projection stabilization of the convective term using an orthogonal subscale decomposition. The accuracy and efficiency of this method compared with residual-based algebraic subgrid scales and orthogonal subscales methods for equal-order interpolation is assessed in this paper. Moreover, when inf–sup stable elements are used, the grad–div stabilization term has been shown to be essential to guarantee accurate solutions. Hence, a study of the influence of such term in the large eddy simulation of turbulent incompressible flows is also performed. Furthermore, a recursive block preconditioning strategy has been considered for the resolution of the problem with an implicit treatment of the projection terms. Two different benchmark tests have been solved: the Taylor–Green Vortex flow with Re=1600, and the Turbulent Channel Flow at Reτ=395 and Reτ=590.

The effect of a sparse grad–div stabilization on control of stationary Navier–Stokes equations

This study presents a reduced sparsity grad–div stabilization for the optimal control of an incompressible stationary flow problem. The effect of the proposed stabilization is analyzed in terms of stability and velocity a priori error. The main difference between this method and the usual grad–div stabilization can be observed in numerical computations. Since the proposed method produces block upper triangular matrices, an advantage can be seen in terms of implementation over the common grad–div stabilization, which produces fully coupled block matrices. The stability and the convergence results are given for both the state and the adjoint state variables once the optimality conditions by using the Lagrange approach are stated. In numerical experiments, solutions of the optimal control problem without any stabilization, and the proposed method are presented in order to compare the effectiveness of the method. The numerical convergence study is also presented.

Optimal bilinear control of eddy current equations with grad–div regularization

AbstractAn optimal bilinear control problem governed by time-harmonic eddy current equations is considered to estimate the electric conductivity of a 3D bounded isotropic domain. The model problem is mainly complicated by the possible presence of non-conducting materials in the domain. We introduce an optimal control approach based on grad-div regularization and divergence penalization. The estimation for the electric conductivity obtained by solving the optimal control problem is allowed to be discontinuous. Here, no higher regularity property can be derived from the corresponding optimality conditions. We analyze the approach and present various numerical results exhibiting its numerical performance

A fast multigrid-based electromagnetic eigensolver for curved metal boundaries on the Yee mesh

For embedded boundary electromagnetics using the Dey–Mittra (Dey and Mittra, 1997) [1] algorithm, a special grad–div matrix constructed in this work allows use of multigrid methods for efficient inversion of Maxwell’s curl–curl matrix. Efficient curl–curl inversions are demonstrated within a shift-and-invert Krylov-subspace eigensolver (open-sourced at [ofortt]https://github.com/bauerca/maxwell[cfortt]) on the spherical cavity and the 9-cell TESLA superconducting accelerator cavity. The accuracy of the Dey–Mittra algorithm is also examined: frequencies converge with second-order error, and surface fields are found to converge with nearly second-order error. In agreement with previous work (Nieter et al., 2009) [2], neglecting some boundary-cut cell faces (as is required in the time domain for numerical stability) reduces frequency convergence to first-order and surface-field convergence to zeroth-order (i.e. surface fields do not converge). Additionally and importantly, neglecting faces can reduce accuracy by an order of magnitude at low resolutions.

On a reduced sparsity stabilization of grad–div type for incompressible flow problems

We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad–div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad–div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator’s matrices are identical to those of grad–div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad–div stabilization, and in all cases, solutions are found to be very similar.

Numerical solution of nonlocal hydrodynamic Drude model for arbitrary shaped nano-plasmonic structures using Nédélec finite elements

Nonlocal material response distinctively changes the optical properties of nano-plasmonic scatterers and waveguides. It is described by the nonlocal hydrodynamic Drude model, which – in frequency domain – is given by a coupled system of equations for the electric field and an additional polarization current of the electron gas modeled analogous to a hydrodynamic flow. Recent attempt to simulate such nonlocal model using the finite difference time domain method encountered difficulties in dealing with the grad–div operator appearing in the governing equation of the hydrodynamic current. Therefore, in these studies the model has been simplified with the curl-free hydrodynamic current approximation; but this causes spurious resonances. In this paper we present a rigorous weak formulation in the Sobolev spaces H(curl) for the electric field and H(div) for the hydrodynamic current, which directly leads to a consistent discretization based on Nédélec’s finite element spaces. Comparisons with the Mie theory results agree well. We also demonstrate the capability of the method to handle any arbitrary shaped scatterer.