Kaibo Hu

Bio: Kaibo Hu is an academic researcher from University of Minnesota. The author has contributed to research in topics: Finite element method & Mathematics. The author has an hindex of 8, co-authored 33 publications receiving 311 citations. Previous affiliations of Kaibo Hu include Peking University & University of Oslo.

Stable finite element methods preserving $$\nabla \cdot \varvec{B}=0$$ź·B=0 exactly for MHD models

Abstract: 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's law, namely $$ abla \cdot \varvec{B}=0$$ź·B=0. In contrast to most existing approaches that eliminate the electrical field variable $$\varvec{E}$$E and give a direct discretization of the magnetic field, our new approach discretizes the electric field $$\varvec{E}$$E by Nedelec type edge elements for $$H(\mathrm {curl})$$H(curl), while the magnetic field $$\varvec{B}$$B by Raviart---Thomas type face elements for $$H(\mathrm {div})$$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 the Picard linearization of the fully nonlinear systems by using the Brezzi theory. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.

Generalized finite element systems for smooth differential forms and Stokes’ problem

Abstract: 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.

Nodal finite element de Rham complexes

Abstract: 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 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 Nedelec and Brezzi–Douglas–Marini finite elements, and the basis functions are more canonical and easier to construct. Our finite elements for $${H}(\mathrm {div})$$ with regularity $$r=2$$ coincide with the nonstandard elements given by Stenberg (Numer Math 115(1):131–139, 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 finite element systems.

Complexes from complexes

Abstract: This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the derived complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincare type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.

Fundamentals Of Differential Geometry

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Towards a Scalable Fully-Implicit Fully-coupled Resistive MHD Formulation with Stabilized FE Methods

Abstract: This paper presents an initial study that is intended to explore the development of a scalable fully-implicit stabilized unstructured finite element (FE) capability for low-Mach-number resistive MHD. The discussion considers the development of the stabilized FE formulation and the underlying fully-coupled preconditioned Newton-Krylov nonlinear iterative solver. To enable robust, scalable and efficient solution of the large-scale sparse linear systems generated by the Newton linearization, fully-coupled algebraic multilevel preconditioners are employed. Verification results demonstrate the expected order-of-acuracy for the stabilized FE discretization of a 2D vector potential form for the steady and transient solution of the resistive MHD system. In addition, this study puts forth a set of challenging prototype problems that include the solution of an MHD Faraday conduction pump, a hydromagnetic Rayleigh-Bernard linear stability calculation, and a magnetic island coalescence problem. Initial results that explore the scaling of the solution methods are presented on up to 4096 processors for problems with up to 64M unknowns on a CrayXT3/4. Additionally, a large-scale proof-of-capability calculation for 1 billion unknowns for the MHD Faraday pump problem on 24,000 cores is presented.

A fully divergence-free finite element method for magnetohydrodynamic equations

Abstract: We propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magne...