An Adaptive P 1 Finite Element Method for Two-Dimensional Maxwell’s Equations

Recently a new numerical approach for two-dimensional Maxwell’s equations based on the Hodge decomposition for divergence-free vector fields was introduced by Brenner et al. In this paper we present an adaptive P ₁ finite element method for two-dimensional Maxwell’s equations that is based on this new approach. The reliability and efficiency of a posteriori error estimators based on the residual and the dual weighted-residual are verified numerically. The performance of the new approach is shown to be competitive with the lowest order edge element of Nédélec’s first family.     

Non-Reflecting Boundary Conditions for Maxwell’s Equations

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.     

Unified Analysis of Leap-Frog Methods for Solving Time-Domain Maxwell’s Equations in Dispersive Media

In this paper, we consider the time dependent Maxwell’s equations resulting from dispersive medium models. First, the stability and Gauss’s law are proved for all three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Then leap-frog mixed finite element methods are developed for these three models. Optimal error estimates are proved for all three models solved by the lowest-order Raviart-Thomas-Nédélec spaces. Extensions to multiple pole dispersive media are presented also. Numerical results confirming the analysis are presented.     

Multigrid Methods for a Mixed Finite Element Method of the Darcy–Forchheimer Model

An efficient nonlinear multigrid method for a mixed finite element method of the Darcy–Forchheimer model is constructed in this paper. A Peaceman–Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.     

Convergence Analysis of Primal–Dual Based Methods for Total Variation Minimization with Finite Element Approximation

We consider a minimization model with total variational regularization, which can be reformulated as a saddle-point problem and then be efficiently solved by the primal–dual method. We utilize the consistent finite element method to discretize the saddle-point reformulation; thus possible jumps of the solution can be captured over some adaptive meshes and a generic domain can be easily treated. Our emphasis is analyzing the convergence of a more general primal–dual scheme with a combination factor for the discretized model. We establish the global convergence and derive the worst-case convergence rate measured by the iteration complexity for this general primal–dual scheme. This analysis is new in the finite element context for the minimization model with total variational regularization under discussion. Furthermore, we propose a prediction–correction scheme based on the general primal–dual scheme, which can significantly relax the step size for the discretization in the time direction. Its global convergence and the worst-case convergence rate are also established. Some preliminary numerical results are reported to verify the rationale of considering the general primal–dual scheme and the primal–dual-based prediction–correction scheme.     

Convergence to Suitable Weak Solutions for a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

In this article, a Galerkin finite element approximation for a class of time–space fractional differential equation is studied, under the assumption that \(u_{tt}, u_{ttt}, u_{2\alpha ,tt}\) are continuous for \(\varOmega \times (0,T]\), but discontinuous at time \(t=0\). In spatial direction, the Galerkin finite element method is presented. And in time direction, a Crank–Nicolson time-stepping is used to approximate the fractional differential term, and the product trapezoidal method is employed to treat the temporal fractional integral term. By using the properties of the fractional Ritz projection and the fractional Ritz–Volterra projection, the convergence analyses of semi-discretization scheme and full discretization scheme are derived separately. Due to the lack of smoothness of the exact solution, the numerical accuracy does not achieve second order convergence in time, which is \(O(k^{3-\beta }+k^{3}t_{n+1}^{-\beta }+k^{3}t_{n+1}^{-\beta -1})\), \(n=0,1,\ldots ,N-1\). But the convergence order in time is shown to be greater than one. Numerical examples are also included to demonstrate the effectiveness of the proposed method.     

A Linearized Local Conservative Mixed Finite Element Method for Poisson–Nernst–Planck Equations

In this paper, a linearized local conservative mixed finite element method is proposed and analyzed for Poisson–Nernst–Planck (PNP) equations, where the mass fluxes and the potential flux are introduced as new vector-valued variables to equations of ionic concentrations (Nernst–Planck equations) and equation of the electrostatic potential (Poisson equation), respectively. These flux variables are crucial to PNP equations on determining the Debye layer and computing the electric current in an accurate fashion. The Raviart–Thomas mixed finite element is employed for the spatial discretization, while the backward Euler scheme with linearization is adopted for the temporal discretization and decoupling nonlinear terms, thus three linear equations are separately solved at each time step. The proposed method is more efficient in practice, and locally preserves the mass conservation. By deriving the boundedness of numerical solutions in certain strong norms, an unconditionally optimal error analysis is obtained for all six unknowns: the concentrations p and n, the mass fluxes \({{\varvec{J}}}_p=\nabla p + p {\varvec{\sigma }}\) and \({{\varvec{J}}}_n=\nabla n - n {\varvec{\sigma }}\), the potential \(\psi \) and the potential flux \({\varvec{\sigma }}= \nabla \psi \) in \(L^{\infty }(L^2)\) norm. Numerical experiments are carried out to demonstrate the efficiency and to validate the convergence theorem of the proposed method.     

Optimal Penalty Parameters for Symmetric Discontinuous Galerkin Discretisations of the Time-Harmonic Maxwell Equations

We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are (i) valid in the pre-asymptotic regime; (ii) solely depend on the geometry and the polynomial order; and (iii) are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order (p=1, 2, 3, 4) hierarchic H(curl)-conforming polynomial basis functions.     

An easy and efficient combination of the Mixed Finite Element Method and the Method of Lines for the resolution of Richards’ Equation

In this work, the Mixed Hybrid Finite Element (MHFE) method is combined with the Method Of Lines (MOL) for an accurate resolution of the Richard's Equation (RE). The combination of these methods is often complicated since hybridization requires a discrete approximation of the time derivative whereas with the MOL, it should remain continuous. In this paper, we use the new mass lumping technique developed in Younes et al. [Younes, A., Ackerer, P., Lehmann, F., 2006. A new mass lumping scheme for the mixed hybrid finite element method. International Journal for Numerical Methods in Engineering 67, pp. 89–107.] for the MHFE method. With this formulation, the MOL is easily implemented and sophisticated time integration packages can be used without significant amount of work.Numerical simulations are performed on both homogeneous and heterogeneous porous media to show the efficiency and robustness of the developed scheme.     

Locally Implicit Time Integration Strategies in a Discontinuous Galerkin Method for Maxwell’s Equations

An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.     

Adaptive Finite Element Approximation for a Constrained Optimal Control Problem via Multi-meshes

In this paper, we study adaptive finite element approximation schemes for a constrained optimal control problem. We derive the equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.     

A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations

In this paper we present a Legendre pseudospectral algorithm based on a tensor product formulation for solving the time-domain Maxwell equations. Our approach starts by conducting an analysis for finding well-posed boundary operators for the Maxwell equations. We then discuss equivalent characteristic boundary conditions for common physical boundary constraints. These theoretical results are then employed to construct a pseudospectral penalty scheme which is asymptotically stable at the semidiscrete level. Numerical computations based on the proposed scheme are also provided for different cases where exact solutions exist. By measuring the differences between the computed and exact solutions, we observe the expected convergence patterns of the scheme. This work is supported by National Science Council grant No. NSC 95-2120-M-001-003.     

Finite Element Computation of KPP Front Speeds in Cellular and Cat’s Eye Flows

We compute the front speeds of the Kolmogorov-Petrovsky-Piskunov (KPP) reactive fronts in two prototypes of periodic incompressible flows (the cellular flows and the cat’s eye flows). The computation is based on adaptive streamline diffusion methods for the advection-diffusion type principal eigenvalue problem associated with the KPP front speeds. In the large amplitude regime, internal layers form in eigenfunctions. Besides recovering known speed growth law for the cellular flow, we found larger growth rates of front speeds in cat’s eye flows due to the presence of open channels, and the front speed slowdown due to either zero Neumann boundary condition at domain boundaries or frequency increase of cat’s eye flows.