Mixed Discretization Research Articles

Multigrid methods for convergent mixed finite difference scheme for Monge–Ampère equation

We propose multigrid methods for convergent mixed finite difference discretization for the two dimensional Monge–Ampere equation. We apply mixed standard 7-point stencil and semi-Lagrangian wide stencil discretization, such that the numerical solution is guaranteed to converge to the viscosity solution of the Monge–Ampere equation. We investigate multigrid methods for two scenarios. The first scenario considers applying standard 7-point stencil discretization on the entire computational domain. We use full approximation scheme with four-directional alternating line smoothers. The second scenario considers the more general mixed stencil discretization and is used for the linearized problem. We propose a coarsening strategy where wide stencil points are set as coarse grid points. Linear interpolation is applied on the entire computational domain. At wide stencil points, injection as the restriction yields a good coarse grid correction. Numerical experiments show that the convergence rates of the proposed multigrid methods are mesh-independent.

Mixed Discretization of the Time-Domain MFIE at Low Frequencies

Solution of the magnetic field integral equation, which is obtained by the classical marching-on-in-time (MOT) scheme, becomes inaccurate when the time-step is large, i.e., under low-frequency excitation. It is shown here that the inaccuracy stems from the classical MOT scheme's failure to predict the correct scaling of the current's Helmholtz components for large time-steps. A recently proposed mixed discretization strategy is used to alleviate the inaccuracy problem by restoring the correct scaling of the current's Helmholtz components under low-frequency excitation.

Finite element exterior calculus with lower-order terms

The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they frequently show up perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is well understood, but that in the vector case is not. In this paper, we first show that, surprisingly, for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further, our results imply many previous results for the scalar problem and thus unify them all under the FEEC framework.

A multiscale flux basis for mortar mixed discretizations of Stokes–Darcy flows

A multiscale flux basis algorithm is developed for the Stokes–Darcy flow problem. The method is based on a non-overlapping domain decomposition algorithm, where the global problem is reduced to a coarse scale mortar interface problem that is solved by an iterative solver. Subdomain solves are required at each interface iteration, so the cost for the method without a multiscale basis can be high when the number of subdomains or the condition number of the interface problem is large. The proposed algorithm involves precomputing a multiscale flux basis, which consists of the flux (or velocity trace) response from each mortar degree of freedom. It is computed by each subdomain independently before the interface iteration begins. The subdomain solves required at each iteration are substituted by a linear combination of the multiscale basis. This may lead to a significant reduction in computational cost since the number of subdomain solves is fixed, depending only on the number of mortar degrees of freedom associated with a subdomain. Several numerical examples are carried out to demonstrate the efficiency of the multiscale flux basis implementation for large-scale Stokes–Darcy problems.

Multidomain-staggered coupling technique for Darcy–Navier Stokes multiphase flow: An application to CO2 geosequestration

This paper introduces a multidomain-staggered technique for coupling multiphase flow in a porous medium, dominated by the Darcy laminar flow, with multiphase flow in a wellbore, dominated by the Navier Stokes viscous, compressible flow. The Darcy flow in the porous medium is formulated using the averaging theory, and the Navier Stokes flow in the wellbore is formulated using the drift-flux model. The governing equations are discretized using a mixed discretization finite element scheme, in which the partition of unity finite element method, the level set method and the standard Galerkin finite element method are combined in an integrated numerical scheme. A multidomain technique is utilized to uncouple the physical system into two subdomains, coupled back by enforcing flow constraints at their interaction boundaries. The resulting system of equations is solved using an iterative staggered technique and a multiple time-stepping scheme. This combination between the multidomain technique and the staggered-multiple time-stepping technique enables the use of different mathematical and numerical formulations for the two subdomains, and facilitates the implementation of a standard finite element computer code. The proposed model is tailored to simulate sequestered CO2 leakage through heterogeneous geological formation layers and abandoned wellbores. A numerical example describing different leakage scenarios is given to demonstrate the computational capability of the model. The numerical results are compared to those obtained from a commercial simulator.

Topology optimization with mixed finite elements on regular grids

Recently, new families of mixed finite elements have been proposed to address the analysis of linear elastic bodies on regular grids adopting a limited number of degrees of freedom per element. A two-dimensional mixed discretization is implemented to formulate an alternative topology optimization problem where stresses play the role of main variables and both compressible and incompressible materials can be dealt with. The structural compliance is computed through the evaluation of the complementary energy, whereas the enforcement of stress constraints is straightforward. Numerical simulations investigate the features of the proposed approach: comparisons with a conventional displacement-based scheme are provided for compressible materials; stress-constrained solutions for structures made of incompressible media are introduced.

Boundary Element Formulation for Numerical Surface Wave Modelling in Poroviscoelastisity

Problems of the poroviscodynamics are considered. Theory of poroviscoelasticity is based on Biot’s equations of fluid saturated porous media under assumption that the skeleton is viscoelastic. Viscoelastic effects of solid skeleton are modeled by mean of elastic-viscoelastic correspondence principle, using such viscoelastic models as a standard linear solid model and model with weakly singular kernel. The fluid is taken as original in Biot’s formulation without viscoelastic effects. Boundary integral equations method is applied to solve three-dimensional boundary-value problems. Boundary-element method with mixed discretization and matched approximation of boundary functions is used. Solution is obtained in Laplace domain, and then Durbin’s algorithm of numerical inversion of Laplace transform is applied to perform solution in time domain. An influence of viscoelastic parameters on dynamic responses is studied. Numerical example of the surface waves modelling is considered.

NCP Function--Based Dual Weighted Residual Error Estimators for Signorini's Problem

In this paper, we consider goal-oriented adaptive finite element methods for Signorini's problem. The basis is a mixed formulation, which is reformulated as nonlinear variational equality using a nonlinear complementarity function. For a general discretization, we derive error identities w.r.t. a possible nonlinear quantity of interest in the displacement as well as in the contact forces, which are included as Lagrange multiplier, using the dual weighted residual method. Afterwards, a numerical approximation of the error identities is introduced. We exemplify the results for a low order mixed discretization of Signorini's problem. The theorectical findings and the numerical approximation scheme are finally substantiated by some numerical examples.

Divergence-conforming discretization for Stokes problem on locally refined meshes using LR B-splines

Abstract To solve the incompressible flow problems using isogeometric analysis, the div-compatible spline spaces were originally introduced by Buffa et al. (2011), and later developed by Evans (2011). In this paper, we extend the div-compatible spline spaces with local refinement capability using Locally Refined (LR) B-splines over rectangular domains. We argue that the spline spaces generated on locally refined meshes will satisfy compatibility provided they span the entire function spaces as governed by Mourrain (2014) dimension formula. We will in this work use the structured refined LR B-splines as introduced by Johannessen et al. (2014). Further, we consider these div-compatible LR B-spline spaces to approximate the velocity and pressure fields in mixed discretization for Stokes problem and a set of standard benchmark tests are performed to show the stability, efficiency and the conservation properties of the discrete velocity fields in adaptive isogeometric analysis.

An Analysis of the Blended Three-Step Backward Differentiation Formula Time-Stepping Scheme for the Navier-Stokes-Type System Related to Soret Convection

In this article, we investigate the stability and convergence of a new class of blended three-step Backward Differentiation Formula (BDF) time-stepping scheme for spatially discretized Navier-Stokes-type system modeling Soret driven convective flows. A Galerkin mixed finite element spatial discretization is assumed, and the temporal discretization is by the implicit blended three-step BDF scheme. The blended BDF scheme is more accurate than the classical second order accurate two-step BDF (BDF2) scheme, yet strongly A-stable. We consider an implicit, linearly extrapolated version of the scheme to improve its efficiency. We present optimal finite element error estimates and prove the scheme is unconditionally stable and convergent. Numerical experiments are presented that compare the scheme to the classical BDF2 scheme.

A computational model for fluid leakage in heterogeneous layered porous media

This paper introduces a new and computationally efficient model for the simulation of non-wetting phase leakage in a rigid heterogeneous layered medium domain constituting layers of different physical properties. Such a leakage exhibits a discontinuity in the saturation field at the interface between layers. The governing field equations are derived based on the averaging theory and solved numerically using a mixed finite element discretization scheme. This scheme entails solving different balance equations using different discretization techniques, which are tailored to accurately simulate the physical behavior of the primary state variables. A discontinuous non-wetting phase saturation–continuous water pressure formulation is adopted. The standard Galerkin finite element method is utilized to discretize the water phase pressure field, and the partition of unity finite element method is utilized to discretize the non-wetting phase saturation field. This mixed discretization scheme leads to a locally conservative system, giving accurate simulation of the saturation jump. The boundary between layers is embedded within the finite elements, alleviating the need to use the typical interface elements, and allowing for the use of structured, geometry-independent and relatively coarse meshes. The accuracy and capability of the proposed model are evaluated by verification and numerical examples covering water, DNAPL and CO2 leakage through layers of different hydraulic properties.

Low-Frequency Scaling of the Standard and Mixed Magnetic Field and Müller Integral Equations

The standard and mixed discretizations for the magnetic field integral equation (MFIE) and the Muller integral equation (MUIE) are investigated in the context of low-frequency (LF) scattering problems involving simply connected scatterers. It is proved that, at low frequencies, the frequency scaling of the nonsolenoidal part of the solution current can be incorrect for the standard discretization. In addition, it is proved that the frequency scaling obtained with the mixed discretization is correct. The reason for this problem in the standard discretization scheme is the absence of exact solenoidal currents in the rotated RWG finite element space. The adoption of the mixed discretization scheme eliminates this problem and leads to a well-conditioned system of linear equations that remains accurate at low frequencies. Numerical results confirm these theoretical predictions and also show that, when the frequency is lowered, a finer and finer mesh is required to keep the accuracy constant with the standard discretization.

Low-Complexity Finite Element Algorithms for the de Rham Complex on Simplices

We combine recently developed finite element algorithms based on Bernstein polynomials [M. Ainsworth, G. Andriamaro, and O. Davydov, SIAM J. Sci. Comput., 33 (2011), pp. 3087--3109; R. C. Kirby, Numer. Math., 117 (2011), pp. 631--652] with the explicit basis construction of the finite element exterior calculus [D. N. Arnold, R. S. Falk, and R. Winther, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1660--1672] to give a family of algorithms for the de Rham complex on simplices that achieves stiffness matrix construction and matrix-free action in optimal complexity. These algorithms are based on realizing the exterior calculus bases as short combinations of Bernstein polynomials. Numerical results confirm optimal scaling of the algorithms and favorable comparison with FEniCS at high polynomial order as well. Additional empirical studies show that very high accuracy is achieved in the mixed discretization of the Poisson equation and the Maxwell eigenvalue problem as the polynomial degree increases.

Numerical modelling of three-phase immiscible flow in heterogeneous porous media with gravitational effects

This paper presents a new numerical formulation for the simulation of immiscible and incompressible three-phase water–gas–oil flows in heterogeneous porous media. We take into account the gravitational effects, both variable permeability and porosity of porous medium, and explicit spatially varying capillary pressure, in the diffusive fluxes, and explicit spatially varying flux functions, in the hyperbolic operator. The new formulation is a sequential time marching fractional-step procedure based in a splitting technique to decouple the equations with mixed discretization techniques for each of the subproblems: convection, diffusion, and pressure–velocity. The system of nonlinear hyperbolic equations that models the convective transport of the fluid phases is approximated by a modified central scheme to take into account the explicit spatially discontinuous flux functions and the effects of spatially variable porosity. This scheme is coupled with a locally conservative mixed finite element formulation for solving parabolic and elliptic problems, associated respectively with the diffusive transport of fluid phases and the pressure–velocity problem. The time discretization of the parabolic problem is performed by means of an implicit backward Euler procedure. The hybrid-mixed formulation reported here is designed to handle discontinuous capillary pressures. The new method is used to numerically investigate the question of existence, and structurally stable, of three-phase flow solutions for immiscible displacements in heterogeneous porous media with gravitational effects. Our findings appear to be consistent with theoretical and experimental results available in the literature.

A computational model for coupled multiphysics processes of CO2 sequestration in fractured porous media

In this paper, a computational model for the simulation of coupled hydromechanical and electrokinetic flow in fractured porous media is introduced. Particular emphasis is placed on modeling CO2 flow in a deformed, fractured geological formation and the associated electrokinetic flow. The governing field equations are derived based on the averaging theory and the double porosity model. They are solved numerically with a mixed discretization scheme, formulated on the basis of the standard Galerkin finite element method, the extended finite element method, the level-set method and the Petrov–Galerkin method. The standard Galerkin method is utilized to discretize the equilibrium and the diffusive dominant field equations, and the extended finite element method, together with the level-set method and the Petrov–Galerkin method, are utilized to discretize the advective dominant field equations. The level-set method is employed to trace the CO2 plume front, and the extended finite element method is employed to model the high gradient in the saturation field front. The proposed mixed discretization scheme leads to a convergent system, giving a stable and effectively mesh-independent model. The accuracy and computational efficiency of the proposed model is evaluated by verification and numerical examples. Effects of the fracture spacing on the CO2 flow and the streaming potential are discussed.

Coupled Electrokinetic–Hydromechanic Model for $$\text{ CO}_{2}$$ Sequestration in Porous Media

In this paper, a computational model for the simulation of coupled electrokinetic and hydromechanical flow in a multiphase domain is introduced. Particular emphasis is placed on modeling $$\text{ CO}_{2}$$ flow in a deformed, unsaturated geologic formation and its associated streaming potential. The governing field equations are derived based on the averaging theory and solved numerically based on a mixed discretization scheme. The standard Galerkin finite element method is utilized to discretize the deformation and the diffusive dominant field equations, and the extended finite element method, together with the level-set method, is utilized to discretize the advective dominant field equations. The level-set method is employed to trace the $$\text{ CO}_{2}$$ plume front, and the extended finite element method is employed to model the high gradient in the saturation field front. This mixed discretization scheme leads to a highly convergent system, giving a stable and effectively mesh-independent model; furthermore, it minimizes the number of degrees of freedom, making the numerical scheme computationally efficient. The capability of the proposed model is evaluated by verification and numerical examples. Effects of the formation stiffness on the $$\text{ CO}_{2}$$ flow and the salinity content on the streaming potential are discussed.

A Space-Time Mixed Galerkin Marching-on-in-Time Scheme for the Time-Domain Combined Field Integral Equation

The time domain combined field integral equation (TD-CFIE), which is constructed from a weighted sum of the time domain electric and magnetic field integral equations (TD-EFIE and TD-MFIE) for analyzing transient scattering from closed perfect electrically conducting bodies, is free from spurious resonances. The standard marching-on-in-time technique for discretizing the TD-CFIE uses Galerkin and collocation schemes in space and time, respectively. Unfortunately, the standard scheme is theoretically not well understood: stability and convergence have been proven for only one class of space-time Galerkin discretizations. Moreover, existing discretization schemes are nonconforming, i.e., the TD-MFIE contribution is tested with divergence conforming functions instead of curl conforming functions. We therefore introduce a novel space-time mixed Galerkin discretization for the TD-CFIE. A family of temporal basis and testing functions with arbitrary order is introduced. It is explained how the corresponding interactions can be computed efficiently by existing collocation-in-time codes. The spatial mixed discretization is made fully conforming and consistent by leveraging both Rao-Wilton-Glisson and Buffa-Christiansen basis functions and by applying the appropriate bi-orthogonalization procedures. The combination of both techniques is essential when high accuracy over a broad frequency band is required.

Resolution of Unsteady Navier-stokes Equations with the C a,b Boundary condition

in this work, we introduce the unsteady incompressible Navier-Stokes equations with a new boundary condition, generalizes the Dirichlet and the Neumann conditions. Then we derive an adequate variational formulation of time-dependent Navier- Stokes equations. We then prove the existence theorem and a uniqueness result. A Mixed finite-element discretization is used to generate the nonlinear system corresponding to the Navier-Stokes equations. The solution of the linearized system is carried out using the GMRES method. In order to evaluate the performance of the method, the numerical results are compared with others coming from commercial code like Adina system.

Discrete spectrum analyses for various mixed discretizations of the Stokes eigenproblem

We conduct discrete spectrum analyses for a selection of mixed discretization schemes for the Stokes eigenproblem. In particular, we consider the MINI element, the Crouzeix---Raviart element, the Marker-and-Cell scheme, the Taylor---Hood element, the $${\mathbf{Q}_{k}/P_{k-1}}$$ element, the divergence-conforming discontinuous Galerkin method, and divergence-conforming B-splines. For each of these schemes, we compare the spectrum for the continuous Stokes problem with the spectrum for the discrete Stokes problem, and we discuss the relationship of eigenvalue errors with solution errors associated with unsteady viscous flow problems.

A posteriori error estimates of higher-order finite elements for frictional contact problems

In this paper, a posteriori estimates are derived for higher-order finite element methods and frictional contact problems. The discretization is based on a mixed approach where the geometrical and frictional constraints are captured by Lagrange multipliers. The use of higher-order polynomials leads to a certain non-conformity in the discretization which requires special attention in the error analysis. As a main result an error estimation is proposed which consists of the dual norm of a residual plus some computable remainder terms. The residual is estimated by well-known a posteriori error estimates for variational equations. The remainder terms represent typical sources resulting from the non-conforming mixed discretization. Numerical experiments confirm the applicability of the a posteriori estimates to adaptive mesh refinements.