• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.237-260
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• 2001

The approximation properties for $L^2$-projection, Raviart-Thomas projection, and inverse inequality have been derived in 3 dimensional space. h-p-mixed finite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed in 3D. Solvability and convergence of the linearized problem have been shown through duality argument and fixed point argument. The analysis is carried out in detail using Raviart-Thomas-Nedelec spaces as an example.

• Journal of the Korean Mathematical Society
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• v.61 no.4
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• pp.659-681
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• 2024

We propose V-cycle multigrid methods for vector field problems arising from the lowest order hexahedral Nédélec finite element. Since the conventional scalar smoothing techniques do not work well for the problems, a new type of smoothing method is necessary. We introduce new smoothers based on substructuring with nonoverlapping domain decomposition methods. We provide the convergence analysis and numerical experiments that support our theory.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.4
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• pp.323-332
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• 2022

This paper presents a numerical study on multigrid algorithms of V-cycle type for problems posed in the Hilbert space H(curl) in three dimensions. The multigrid methods are designed for discrete problems originated from the discretization using the hexahedral Nédélec edge element of the lowest-order. Our suggested methods are associated with smoothers constructed by substructuring based on domain decomposition methods of nonoverlapping type. Numerical experiments to demonstrate the robustness and the effectiveness of the suggested algorithms are also provided.