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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.

In this paper, we derive an a posteriori error estimator for the weak Galerkin least-squares (WG-LS) method applied to the reaction–diffusion equation. We show that this estimator is both reliable and efficient, allowing it to be used for adaptive refinement. Due to the flexibility of the WG-LS discretization, we are able to design a simple and straightforward refinement scheme that is applicable to any shape regular polygonal mesh. Finally, we present numerical experiments that confirm the effectiveness of the estimator, and demonstrate the robustness and efficiency of the proposed adaptive WG-LS approach.

This paper introduces a weak Galerkin (WG) finite element method for the Navier–Stokes equations in the primal velocity–pressure formulation. Optimal-order error estimates are established for the corresponding numerical approximations. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular. Numerical experiments are presented to support the theoretical results.

Vector-potential formulations are attractive for electromagnetic problems in two dimensions, since they reduce both the number and complexity of equations, particularly in coupled systems, such as magnetohydrodynamics (MHD). In this paper, we consider the finite-element formulation of a vector-potential model of two-dimensional resistive MHD. Existence and uniqueness are considered separately for the continuum nonlinear equations and the discretized and linearized form that arises from Newton’s method applied to a modified system. Under some conditions, we prove that the solutions of the original and modified weak forms are the same, allowing us to prove convergence of the discretization and well-posedness of the nonlinear iteration near a solution.

In this work, we show the underlying mathematical structure of mixed-dimensional models arising from the composition of graphs and continuous domains. Such models are becoming popular in applications, in particular, to model the human vasculature. We first discuss the model equations in the strong form, which describes the conservation of mass and Darcy's law in the continuum and network as well as the coupling between them. By introducing proper scaling, we propose a weak form that avoids degeneracy. Well-posedness of the weak form is shown through standard Babu\v ska--Brezzi theory. We also develop the mixed formulation finite-element method and prove its well-posedness. A mass-lumping technique is introduced to derive the two-point flux approximation (TPFA) type discretization as well, due to its importance in applications. Based on the Babu\v ska--Brezzi theory, error estimates can be obtained for both the finite-element scheme and the TPFA scheme. We also discuss efficient linear solvers for discrete problems. Finally, we present some numerical examples to verify the theoretical results and demonstrate the robustness of our proposed discretization schemes. publishedVersion

Maxwell's equations are a system of partial differential equations that govern the laws of electromagnetic induction. We study a mimetic finite-difference (MFD) discretization of the equations which preserves important underlying physical properties. We show that, after mass-lumping and appropriate scaling, the MFD discretization is equivalent to a structure-preserving finite-element (FE) scheme. This allows for a transparent analysis of the MFD method using the FE framework, and provides an avenue for the construction of efficient and robust linear solvers for the discretized system. In particular, block preconditioners designed for FE formulations can be applied to the MFD system in a straightforward fashion. We present numerical tests which verify the accuracy of the MFD scheme and confirm the robustness of the preconditioners.

This work introduces nodal auxiliary space preconditioners for discretizations of mixed-dimensional partial differential equations. We first consider the continuous setting and generalize the regular decomposition to this setting. With the use of conforming mixed finite element spaces, we then expand these results to the discrete case and obtain a decomposition in terms of nodal Lagrange elements. In turn, nodal preconditioners are proposed analogous to the auxiliary space preconditioners of Hiptmair and Xu [SIAM J. Numer. Anal., 45 (2007), pp. 2483--2509]. Numerical experiments show the performance of this preconditioner in the context of flow in fractured porous media. publishedVersion

We develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure without special treatment. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.

A Newton solver for equations modeling drift-diffusion and electrokinetic phenomena is investigated. For drift-diffusion problems, modeled by the nonlinear Poisson--Nernst--Planck (PNP) equations, the linearization of the model equations is shown to be well-posed. Furthermore, a fast solver for the linearized PNP and electrokinetic equations is proposed and numerically demonstrated to be effective on some physically motivated benchmarks. This work builds on a formulation of the PNP and electrokinetic equations that is investigated in [M. S. Metti, J. Xu, and C. Liu, J. Comput. Phys., 306 (2016), pp. 1--18] and shown to have some favorable stability properties.

A stable finite element scheme that avoids pressure oscillations for a three-field Biot's model in poroelasticity is considered. The involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: Crouzeix-Raviart finite elements for the displacements, lowest order Raviart-Thomas-Nedelecźelements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass-lumping technique is introduced for the Raviart-Thomas-Nedelecźelements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost. We show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass-lumping. Finally, numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the Raviart-Thomas-Nedelecźelements is used.

In this work, we consider two discretizations of the three-field formulation of Biot’s consolidation problem. They employ the lowest-order mixed finite elements for the flow (Raviart-Thomas-Nédélec elements for the Darcy velocity and piecewise constants for the pressure) and are stable with respect to the physical parameters. The difference is in the mechanics: one of the discretizations uses Crouzeix-Raviart nonconforming linear elements; the other is based on piecewise linear elements stabilized by using face bubbles, which are subsequently eliminated. The numerical solutions obtained from these discretizations satisfy mass conservation: the former directly and the latter after a simple postprocessing.

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems., Comment: 27 pages

We consider finite element discretizations of the Biot’s consolidation model in poroelasticity with MINI and stabilized P1–P1 elements. We analyze the convergence of the fully discrete model based on spatial discretization with these types of finite elements and implicit Euler method in time. We also address the issue related to the presence of non-physical oscillations in the pressure approximation for low permeabilities and/or small time steps. We show that even in 1D a Stokes-stable finite element pair fails to provide a monotone discretization for the pressure in such regimes. We then introduce a stabilization term which removes the oscillations. We present numerical results confirming the monotone behavior of the stabilized schemes.

A new consistent, spatially adaptive, smoothed particle hydrodynamics (SPH) method for Fluid-Structure Interactions (FSI) is presented. The method combines several attributes that have not been simultaneously satisfied by other SPH methods. Specifically, it is second-order convergent; it allows for resolutions spatially adapted with moving (translating and rotating) boundaries of arbitrary geometries; and, it accelerates the FSI solution as the adaptive approach leads to fewer degrees of freedom without sacrificing accuracy. The key ingredients in the method are a consistent discretization of differential operators, a \textit{posteriori} error estimator/distance-based criterion of adaptivity, and a particle-shifting technique. The method is applied in simulating six different flows or FSI problems. The new method's convergence, accuracy, and efficiency attributes are assessed by comparing the results it produces with analytical, finite element, and consistent SPH uniform high-resolution solutions as well as experimental data., 27 pages, 17figures

A robust and fast solver for the fractional differential equation (FDEs) involving the Riesz fractional derivative is developed using an adaptive finite element method on non-uniform meshes. It is based on the utilization of hierarchical matrices ($\mathcal{H}$-Matrices) for the representation of the stiffness matrix resulting from the finite element discretization of the FDEs. We employ a geometric multigrid method for the solution of the algebraic system of equations. We combine it with an adaptive algorithm based on a posteriori error estimation to deal with general-type singularities arising in the solution of the FDEs. Through various test examples we demonstrate the efficiency of the method and the high-accuracy of the numerical solution even in the presence of singularities. The proposed technique has been verified effectively through fundamental examples including Riesz, Left/Right Riemann-Liouville fractional derivative and, furthermore, it can be readily extended to more general fractional differential equations with different boundary conditions and low-order terms. To the best of our knowledge, there are currently no other methods for FDEs that resolve singularities accurately at linear complexity as the one we propose here.

In this paper, we design robust and efficient linear solvers for the numerical approximation of solutions to Maxwell's equations with dissipative boundary conditions. We consider a structure-preserving finite-element approximation with standard Nedelec--Raviart--Thomas elements in space and a Crank--Nicolson scheme in time to approximate the electric and magnetic fields. We focus on two types of block preconditioners. The first type is based on the well-posedness results of the discrete problem. The second uses an exact block factorization of the linear system, for which the structure-preserving discretization yields sparse Schur complements. We prove robustness and optimality of these block preconditioners, and provide supporting numerical tests., Comment: 18 pages, 2 figures

In this paper, we concern with a new boundary penalty finite element method for the one-dimensional time-dependent diffusion equation subject to the specification of mass. Instead of solving the diffusion equation directly, we will use the finite element method to solve a boundary penalty problem which is derived from the diffusion equation and analyse the error estimate of the semi-discrete and fully discrete approximations when the penalty parameter e is sufficiently small. The numerical experiments show that our method is very effective and supports our analysis.

In this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified N\'ed\'elec-Raviart-Thomas finite element methods for model problems in $\mathbf{H}(\operatorname{\mathbf{curl}})$ and $H(\operatorname{div})$. This provides a simple and transparent way to analyze such mimetic finite difference discretizations using the well-known results from finite element theory. The finite element framework that we develop is also crucial for the design of efficient multigrid methods for mimetic finite difference discretizations, since it allows us to use canonical inter-grid transfer operators arising from the finite element framework. We provide special Local Fourier Analysis and numerical results to demonstrate the efficiency of such multigrid methods., Comment: submitted