This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss law, namely ∇·B = 0. In contrast to most existing approaches that eliminate the electrical field variable E and give a direct discretization of the magnetic field, our new approach discretizes the electric field E by Nédélec type edge elements for H(curl), while the magnetic field B by Raviart-Thomas type face elements for H(div). As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level.For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for both the Picard and Newton linearization of the fully nonlinear systems by using the Brezzi theory for both the continuous and discrete cases. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.
We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of Finite Element Systems, and the examples include conforming mixed finite elements for Stokes' equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.
We construct 2D and 3D finite element de Rham sequences of arbitrary polynomial degrees with extra smoothness. Some of these elements have nodal degrees of freedom (DoFs) and can be considered as generalisations of scalar Hermite and Lagrange elements. Using the nodal values, the number of global degrees of freedom is reduced compared with the classical Nédélec and Brezzi-Douglas-Marini (BDM) finite elements, and the basis functions are more canonical and easier to construct. Our finite elements for H(div) with regularity r = 2 coincide with the nonstandard elements given by Stenberg (Numer Math 115 (1): [131][132][133][134][135][136][137][138][139] 2010). We show how regularity decreases in the finite element complexes, so that they branch into known complexes. The standard de Rham complexes of Whitney forms and their higher order version can be regarded as the family with the lowest regularity. The construction of the new families is motivated by the finite element systems. * *
ABSTRACT. In this paper, we develop two classes of robust preconditioners for the structurepreserving discretization of the incompressible magnetohydrodynamics (MHD) system. By studying the well-posedness of the discrete system, we design block preconditioners for them and carry out rigorous analysis on their performance. We prove that such preconditioners are robust with respect to most physical and discretization parameters. In our proof, we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters, which are usually difficult to choose in practice. This new technique is not only applicable to the MHD system, but also to other problems. Moreover, we prove that Krylov iterative methods with our preconditioners preserve the divergence-free condition exactly, which complements the structure-preserving discretization. Another feature is that we can directly generalize this technique to other discretizations of the MHD system. We also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners.
In this paper, we develop a class of mixed finite element scheme for stationary magnetohydrodynamics (MHD) models, using magnetic field B and current density j as the discretization variables. We show that the Gauss's law for the magnetic field, namely ∇·B = 0, and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for H h (div), we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of Picard iterations and finite element methods under some conditions.2010 Mathematics Subject Classification. Primary 65N30, 65N12.
We discuss a class of magnetic-electric fields based finite element schemes for stationary magnetohydrodynamics (MHD) systems with two types of boundary conditions. We establish a key L 3 estimate for divergence-free finite element functions for a new type of boundary conditions. With this estimate and a similar one in [21], we rigorously prove the convergence of Picard iterations and the finite element schemes with weak regularity assumptions. These results demonstrate the convergence of the finite element methods for singular solutions.2000 Mathematics Subject Classification. Primary: 65N30, 76W05.Key words and phrases. magnetohydrodynamics, finite element method, structure-preserving, de Rham complex. The title of this article in its original version (see arXiv:1711.11330) is "Magnetic-Electric Formulations for Stationary Magnetohydrodynamics Models".
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