Mixed Finite Element Discretization Research Articles

Distributed Lagrange multiplier/fictitious domain finite element method for Stokes/parabolic interface problems with jump coefficients

In this paper, the distributed Lagrange multiplier/fictitious domain (DLM/FD)–mixed finite element method is developed and analyzed for a type of linearized fluid-structure interaction problem–the transient Stokes/parabolic interface problem with jump coefficients. Stability and optimal convergence properties are obtained for both semi- and fully discrete mixed finite element scheme. Numerical experiments are carried out to validate the theoretical results. Similar DLM/FD method can be developed to tackle FSI problems with different compressibility for the fluid and the structure in the future.

Mixed finite element discretizations of acoustic Helmholtz problems with high wavenumbers

We study the acoustic Helmholtz equation with impedance boundary conditions formulated in terms of velocity, and analyze the stability and convergence properties of lowest-order Raviart-Thomas finite element discretizations. We focus on the high-wavenumber regime, where such discretizations suffer from the so-called “pollution effect”, and lack stability unless the mesh is sufficiently refined. We provide wavenumber-explicit mesh refinement conditions to ensure the well-posedness and stability of discrete scheme, as well as wavenumber-explicit error estimates. Our key result is that the condition “$$k^2 h$$ is sufficiently small”, where k and h respectively denote the wavenumber and the mesh size, is sufficient to ensure the stability of the scheme. We also present numerical experiments that illustrate the theory and show that the derived stability condition is actually necessary.

Auxiliary space preconditioning for mixed finite element discretizations of Richards’ equation

We propose an auxiliary space method for the solution of the indefinite problem arising from mixed method finite element discretizations of scalar elliptic problems. The proposed technique uses conforming elements as an auxiliary space and utilizes special interpolation operators for the transfer of residuals and corrections between the spaces. We show that the corresponding method provides optimal solver for the indefinite problem by only solving symmetric and positive definite auxiliary problems. We apply this preconditioner to the mixed form discretization of Richards’ equation linearized with the L-scheme. We provide numerical tests validating the theoretical estimates.

A Variable Eddington Factor method for the 1-D grey radiative transfer equations with discontinuous Galerkin and mixed finite-element spatial differencing

The purpose of this paper is to present a Variable Eddington Factor (VEF) method for the 1-D grey radiative transfer equations that uses a lumped linear discontinuous Galerkin spatial discretization for the Snequations together with a constant-linear mixed finite-element discretization for the VEF moment and material temperature equations. The use of independent discretizations can be particularly useful for multiphysics applications such as radiation-hydrodynamics. The VEF method is quite old, but to our knowledge, this particular combination of differencing schemes has not been previously investigated for radiative transfer. We define the scheme and present computational results. As expected, the scheme exhibits second-order accuracy for the directionally-integrated intensity and material temperature, and behaves well in the thick diffusion limit. An important focus of this study is the treatment of the strong temperature dependence of the opacities and the spatial dependence of the opacities within each cell, which are not explicitly defined by the basic discretization schemes.

Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale $$H(\mathrm {div})$$ problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this $$H(\mathrm {div})$$ problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart–Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale $$H(\mathrm {div})$$ system is bounded by $$C(1+\frac{H^2}{{\hat{\delta }}^2})(1+\log ^4(\frac{H}{h}))$$ , where $$\hat{\delta }$$ denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.

Local Fourier analysis for mixed finite-element methods for the Stokes equations

In this paper, we develop a local Fourier analysis of multigrid methods based on block-structured relaxation schemes for stable and stabilized mixed finite-element discretizations of the Stokes equations, to analyze their convergence behavior. Three relaxation schemes are considered: distributive, Braess-Sarazin, and Uzawa relaxation. From this analysis, parameters that minimize the local Fourier analysis smoothing factor are proposed for the stabilized methods with distributive and Braess-Sarazin relaxation. Considering the failure of the local Fourier analysis smoothing factor in predicting the true two-grid convergence factor for the stable discretization, we numerically optimize the two-grid convergence predicted by local Fourier analysis in this case. We also compare the efficiency of the presented algorithms with variants using inexact solvers. Finally, some numerical experiments are presented to validate the two-grid and multigrid convergence factors.

Fictitious domain finite element method for Stokes/elliptic interface problems with jump coefficients

In this paper, the distributed Lagrange multiplier/fictitious domain (DLM/FD) finite element method is studied for a generic Stokes/elliptic interface problem with jump coefficients which belongs to a type of linearized stationary fluid–structure interaction problem. A mixed finite element discretization is developed for the proposed DLM/FD method for Stokes/elliptic interface problem and analyzed on the aspects of well-posedness, stability and optimal convergence. Numerical experiments are carried out and the theoretical error estimates of DLM/FD finite element method are validated. • Distributed Lagrange multiplier/fictitious domain finite element method. • Stokes/elliptic interface problem with jump coefficients. • A linearized steady state fluid–structure interaction problem. • New DLM/FD formulation for the incompressible fluid and compressible structure interaction. • Well-posedness, stability and convergence are analyzed for mixed finite elements.

A Relaxed Physical Factorization Preconditioner for Mixed Finite Element Coupled Poromechanics

In this paper, we introduce a relaxed physical factorization (RPF) preconditioner for the efficient iterative solution of the linearized algebraic system arising from the mixed finite element discretization of coupled poromechanics equations. The preconditioner is obtained by using a proper factorization of the $3\times3$ block matrix and setting a relaxation parameter $\alpha$. The preconditioner is inspired by the relaxed dimensional factorization introduced by Benzi et al. [J. Comput. Phys., 230 (2011), pp. 6185--6202; Comput. Methods Appl. Mech. Engrg., 300 (2016), pp. 129--145]. A stable algorithm is advanced to compute the optimal value of $\alpha$, along with a lower bound to control the possible ill-conditioning of the $\alpha$ dependent inner blocks. Numerical experiments in both theoretical benchmarks and real-world applications are presented and discussed to investigate the RPF properties, performance, and robustness.

Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity

We analyze a fully discrete numerical scheme for the model describing two-phase immiscible flow in porous media with dynamic effects in the capillary pressure. We employ the Euler implicit method for the time discretization. The spatial discretization is based on the mixed finite element method (MFEM). Specifically, the lowest order Raviart–Thomas elements are applied. In this paper, the error estimates for the saturation, fluxes and phase pressures in L ∞ ( 0 , T ; L 2 ( Ω ) ) are derived for the temporal and spatial discretization to show the convergence of the scheme. Finally, we present some numerical results to support the theoretical findings.

Mixed finite element analysis for generalized Darcy–Forchheimer model in porous media

We considered mixed finite element discretization of generalized Darcy–Forchheimer model in a two or three dimensional porous domain, using a piecewise constant and continuous piecewise linear finite elements. The existence, uniqueness, stability and convergence of the discrete solution are established. The error estimates for the velocity and pressure in L m and L m + 1 m respectively are obtained for any m ∈ ( 1 , 2 ] . Numerical examples are also presented to confirm the rate of convergence.

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L2 projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.

Numerical Analysis for Time‐Dependent Electromagnetic Field Control

AbstractAn optimal control problem of the full time‐dependent Maxwell equations and its numerical approximation are presented. A mixed finite element discretization in space and a Crank‐Nicolson scheme for the time‐dependent Maxwell equation lead to a discretized optimal control problem. A numerical example illustrates our approach.

Preconditioned Krylov subspace methods for solving radiative transfer problems with scattering and reflection

Two Krylov subspace methods, the GMRES and the BiCGSTAB, are analyzed for solving the linear systems arising from the mixed finite element discretization of the discrete ordinates radiative transfer equation. To increase their convergence rate and stability, the Jacobi and block Jacobi methods are used as preconditioners for both Krylov subspace methods. Numerical experiments, designed to test the effectiveness of the (preconditioned) GMRES and the BiCGSTAB, are performed on various radiative transfer problems: (i) transparent, (ii) absorption dominant, (iii) scattering dominant, and (iv) with specular reflection. It is observed that the BiCGSTAB is superior to the GMRES, with lower iteration counts, solving times, and memory consumption. In particular, the BiCGSTAB preconditioned by the block Jacobi method performed best amongst the set of other solvers. To better understand the discrete systems for radiative problems (i) to (iv), an eigenvalue spectrum analysis has also been performed. It revealed that the linear system conditioning deteriorates for scattering media problems in comparison to absorbing or transparent media problems. This conditioning further deteriorates when reflection is involved.

Multipoint flux mixed finite element methods for slightly compressible flow in porous media

In this paper, we consider multipoint flux mixed finite element discretizations for slightly compressible Darcy flow in porous media. The methods are formulated on general meshes composed of triangles, quadrilaterals, tetrahedra or hexahedra. An inexact Newton method that allows for local velocity elimination is proposed for the solution of the nonlinear fully discrete scheme. We derive optimal error estimates for both the scalar and vector unknowns in the semidiscrete formulation. Numerical examples illustrate the convergence behavior of the methods, and their performance on test problems including permeability coefficients with increasing heterogeneity.

Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q11/Q01 × Q10). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h2 + τ) for u in H1‐norm and p = ∇u in L2‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.

Simultaneous finite element analysis of circuit-integrated piezoelectric energy harvesting from fluid-structure interaction

Abstract Flow-driven piezoelectric energy harvesting is a strongly coupled multiphysics phenomenon that involves complex three-way interaction between the fluid flow, the electromechanical effect of the piezoelectric material mounted on a deformable substrate structure and the controlling electrical circuit. High fidelity computational solution approaches are essential for the analysis of flow-driven energy harvesters in order to capture the main physical aspects of the coupled problem and to accurately predict the power output of a harvester. While there are some phenomenological and numerical models for flow-driven harvesters reported in the literature, a fully three-dimensional strongly coupled model has not yet been developed, especially in the context of flow-driven energy harvesting. The weighted residuals method is applied to establish a mixed integral equation describing the incompressible Newtonian flow, elastic substrate structure, piezoelectric patch, equipotential electrode and attached electric circuit that form the multiphysics fluid-structure interaction problem. A monolithic numerical solution method is derived that provides consistent and simultaneous solution to all physical fields as well as to fluid mesh deformation. The approximate solution is based on a mixed space-time finite element discretization with static condensation of the auxiliary fields. The discontinuous Galerkin method is utilized for integrating the monolithic model in time. The proposed solution scheme is illustrated in the example of a lid driven cavity with a flexible piezoelectric bottom wall, demonstrating quantification of the amount of electrical energy extractable from fluid flow by means of a piezoelectric harvester device. The results indicate that in order to make reliable predictions on the power output under varying operational states, the realization of strong multiphysics coupling is required for the mathematical model as well as the numerical solution scheme to capture the characteristics of flow-driven energy harvesters.

A robust optimal preconditioner for the mixed finite element discretization of elliptic optimal control problems

SummaryIn this paper, we consider the efficient solving of the resulting algebraic system for elliptic optimal control problems with mixed finite element discretization. We propose a block‐diagonal preconditioner for the symmetric and indefinite algebraic system solved with minimum residual method, which is proved to be robust and optimal with respect to both the mesh size and the regularization parameter. The block‐diagonal preconditioner is constructed based on an isomorphism between appropriately chosen solution space and its dual for a general control problem with both state and gradient state observations in the objective functional. Numerical experiments confirm the efficiency of our proposed preconditioner.

Variable Eddington Factor Method for the SN Equations with Lumped Discontinuous Galerkin Spatial Discretization Coupled to a Drift-Diffusion Acceleration Equation with Mixed Finite-Element Discretization

ABSTRACTWe apply the Variable Eddington Factor (VEF) acceleration method to the 1-D, one-group SN equations with a Lumped Linear Discontinuous Galerkin (LLDG) spatial discretization in conjunction with a drift-diffusion acceleration equation discretized with the constant-linear Mixed Finite-Element Method (MFEM). While the VEF method is quite old, its application to this particular combination of SN and drift-diffusion discretizations is new. In addition to demonstrating the basic properties of our method, we investigate slope reconstruction techniques for improving the consistency between the SN and drift-diffusion solutions.

Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties

Abstract This paper introduces a nonlinear multigrid solver for mixed finite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The AMGe coarse spaces with approximation properties used in this work enable us to overcome the difficulties in evaluating the nonlinear coarse operators and the degradation in convergence rates that characterized previous attempts to extend FAS to algebraic multilevel hierarchies on general unstructured grids. Specifically, the AMGe technique employed in this paper allows to derive stable and accurate coarse discretizations on general unstructured grids for a large class of nonlinear partial differential equations, including saddle point problems. The approximation properties of the coarse spaces ensure that our FAS approach for general unstructured meshes leads to optimal mesh-independent convergence rates similar to those achieved by geometric FAS on a nested hierarchy of refined meshes. In the numerical results, Newton’s method and Picard iterations with state-of-the-art inner linear solvers are compared to our FAS algorithm for the solution of a nonlinear saddle point problem arising from porous media flow applications. Our approach outperforms – both in terms of number of iterations and computational time – traditional methods in all the experiments.

Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law

This article is the second of a series where we develop and analyze structure-preserving finite element discretizations for the time-dependent 2D Maxwell system with long-time stability properties, and propose a charge-conserving deposition scheme to extend the stability properties in the case where the current source is provided by a particle method. The schemes proposed here derive from a previous study where a generalized commuting diagram was identified as an abstract compatibility criterion in the design of stable schemes for the Maxwell system alone, and applied to build a series of conforming and non-conforming schemes in the 3D case. Here the theory is extended to account for approximate sources, and specific charge-conserving schemes are provided for the 2D case. In this second article we study two schemes which include a strong discretization of the Faraday law. The first one is based on a standard conforming mixed finite element discretization and the long-time stability is ensured by the natural $L^2$ projection for the current, also standard. The second one is a new non-conforming variant where the numerical fields are sought in fully discontinuous spaces. In this 2D setting it is shown that the associated discrete curl operator coincides with that of a classical DG formulation with centered fluxes, and our analysis shows that a non-standard current approximation operator must be used to yield a charge-conserving scheme with long-time stability properties, while retaining the local nature of $L^2$ projections in discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.