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Title: Remarks on a paper by Gavrilov: Grad–Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications
Award ID(s):
1911413
NSF-PAR ID:
10168824
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Geometric and Functional Analysis
Volume:
29
Issue:
6
ISSN:
1016-443X
Page Range / eLocation ID:
1773 to 1793
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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  2. Abstract We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming hybridizable discontinuous Galerkin schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning technique (Xu (1994) The Auxiliary Space Method and Optimal Multigrid Preconditioning Techniques for Unstructured Grids, vol. 56. International GAMM-Workshop on Multi-level Methods (Meisdorf), pp. 215–235). A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size. 
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