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In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank–Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a second-order decoupling projection method of the Van Kan type [Van Kan, SIAM J. Sci. Statist. Comput.7 (1986) 870–891] in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Error estimates are proved in the discrete L∞(0, T; L2) norm for the proposed decoupled finite element projection scheme. Numerical examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The quad-curl term is an essential part of the resistive magnetohydrodynamic equation and the fourth-order inverse electromagnetic scattering problem, which are both of great significance in science and engineering. It is desirable to develop efficient and practical numerical methods for the quad-curl problem. In this paper we first present some new regularity results for the quad-curl problem on Lipschitz polyhedron domains, and then propose a mixed finite element method for solving the quad-curl problem. With a novel discrete Sobolev imbedding inequality for the piecewise polynomials, we obtain stability results and derive error estimates based on a relatively low-regularity assumption of the exact solution. [ABSTRACT FROM AUTHOR]

It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method, we analyze and solve a two dimensional transmission problem describing elastic waves inside a bounded and closed elastic obstacle and acoustic waves outside it. We are mainly interested in analyzing the DtN boundary condition of the Helmholtz equation in order to establish the well-posedness results of the approximated variational equation, and further derive a priori error estimates involving effects of both the finite element discretization and the truncation of DtN map. Finally, some numerical results are presented to illustrate the accuracy of the numerical scheme. [ABSTRACT FROM AUTHOR]

Abstract In the finite element and boundary element coupling method, the Stekhlov–Poincaré mapping is often adopted at the coupling boundary, i.e., the unknown Neumann data are represented by the unknown Dirichlet data on the same boundary. This coupling method (either by a convolution integral, or by a Fourier series) is referred as the conventional method. We propose an alternative method, where the unknown Dirichlet data at the boundary are represented by the unknown solution value on an interior curve or surface, i.e., a Dirichlet-to-Dirichlet (D-to-D) mapping is employed instead of a Dirichlet-to-Neumann (D-to-N) mapping. The convergence of the method is shown to be of optimal order. 2D and 3D numerical experiments are performed to demonstrate the accuracy of the proposed method, comparing to the two conventional D-to-N coupling methods. [ABSTRACT FROM AUTHOR]

In this paper, we study a hybridizable discontinuous Galerkin (HDG) method for the Maxwell operator. The only global unknowns are defined on the inter-element boundaries, and the numerical solutions are obtained by using discontinuous polynomial approximations. The error analysis is based on a mixed curl-curl formulation for the Maxwell equations. Theoretical results are obtained under a more general regularity requirement. In particular for the low regularity case, special treatment is applied to approximate data on the boundary. The HDG method is shown to be stable and convergence in an optimal order for both high and low regularity cases. Numerical experiments with both smooth and singular analytical solutions are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we investigate numerical solutions of the two-dimensional shallow water wave using a fully nonlinear Green-Naghdi model with an improved dispersive effect. For numerics, the Green-Naghdi model is rewritten into a formulation coupling a pseudo-conservative system and a set of pseudo-elliptic equations. Since the pseudo-conservative system is no longer hyperbolic and its Riemann problem can only be approximately solved, we consider the utilization of the central discontinuous Galerkin method, which possesses an important feature of the needlessness of Riemann solvers. Meanwhile, the stationary elliptic part will be solved using the finite element method. Both the well-balanced and the positivity-preserving features, which are highly desirable in the simulation of the shallow water wave, will be embedded into the proposed numerical scheme. The accuracy and efficiency of the numerical model and method will be illustrated through numerical tests. • A 2D modified Green-Naghdi model with enhanced dispersive property is derived. • A 2D numerical model is presented to reformulate the modified Green-Naghdi model. • A positivity-preserving well balanced CDG-FE method is presented for the model. • The presented method is a high order Galerkin scheme and free of Riemann solver. • A wide range of numerical tests demonstrate the numerical model and scheme. [ABSTRACT FROM AUTHOR]

This paper is concerned with a variational approach solving the two-dimensional acoustic transmission problems. The original problem is reduced to an equivalent nonlocal boundary value problem by introducing an exact Dirichlet-to-Neumann (DtN) mapping in terms of Fourier expansion series on an artificial boundary. Uniqueness and existence of solutions in appropriate Sobolev spaces are established for the corresponding variational problem and its modification due to the truncation of DtN mapping. A priori error estimates containing the effects of both element meshsize and truncation order of series for the finite element approximation are derived. Numerical experiments are also presented to illustrate the efficiency and accuracy of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a temporally second-order accurate numerical scheme for the Cahn–Hilliard-Magnetohydrodynamics system of equations. The scheme is based on a modified Crank–Nicolson-type approximation for the time discretization and a mixed finite element method for the spatial discretization. The modified Crank–Nicolson approximation enables us to carry out the mass conservation and the energy stability analysis. Error estimates are derived for the phase field in the L τ ∞ (0 , T ; H 1) norm, and for the velocity and the magnetic fields in the L τ ∞ (0 , T ; L 2) norm, respectively. Numerical examples are presented to validate the theoretical results of the proposed scheme. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a one-dimensional fully nonlinear weakly dispersive Green–Naghdi model for shallow water waves over variable bottom topographies. Such model describes a large spectrum of shallow water waves, and it is thus of great importance to design accurate and robust numerical methods for solving it. The governing equations contain mixed spatial and temporal derivatives of the unknowns. They also have still-water stationary solutions which should be preserved in stable numerical simulations. In our numerical approach, we first reformulate the Green–Naghdi equations into balance laws coupled with an elliptic equation. We then propose a family of high order numerical methods which discretize the balance laws with well-balanced central discontinuous Galerkin methods and the elliptic part with continuous finite element methods. Linear dispersion analysis for both the (reformulated) Green–Naghdi system and versions of the proposed numerical scheme is performed when the bottom topography is flat. Numerical tests are presented to illustrate the accuracy and stability of the proposed schemes as well as the capability of the Green–Naghdi model to describe a wide range of shallow water wave phenomena. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in and - norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results. [Copyright &y& Elsevier]

Abstract: The interior transmission problem (ITP) is a boundary value problem arising in inverse scattering theory, and it has important applications in qualitative methods. In this paper, we propose a coupled boundary element method (BEM) and a finite element method (FEM) for the ITP in two dimensions. The coupling procedure is realized by applying the direct boundary integral equation method to define the so-called Dirichlet-to-Neumann (DtN) mappings. We show the existence of the solution to the ITP for the anisotropic medium. Numerical results are provided to illustrate the accuracy of the coupling method. [Copyright &y& Elsevier]