• Corpus ID: 232478906

Analysis of injection operators in multigrid solvers for hybridized discontinuous Galerkin methods

@article{Lu2021AnalysisOI,
  title={Analysis of injection operators in multigrid solvers for hybridized discontinuous Galerkin methods},
  author={Peipei Lu and Andreas Rupp and Guido Kanschat},
  journal={ArXiv},
  year={2021},
  volume={abs/2104.00118}
}
Uniform convergence of the geometric multigrid V-cycle is proven for HDG methods with a new set of assumptions on the injection operators from coarser to finer meshes. The scheme involves standard smoothers and local solvers which are bounded, convergent, and consistent. Elliptic regularity is used in the proofs. The new assumptions admit injection operators local to a single coarse grid cell. Examples for admissible injection operators are given. The analysis applies to the hybridized local… 

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SHOWING 1-10 OF 16 REFERENCES
Homogeneous multigrid for embedded discontinuous Galerkin methods
TLDR
A homogeneous multigrid method that uses the same embedded discontinuous Galerkin (EDG) discretization scheme for Poisson’s equation on all levels and proves optimal convergence of the method under the assumption of elliptic regularity is introduced.
Multigrid for an HDG method
TLDR
It is proved that a non-nested multigrid V-cycle, with a single smoothing step per level, converges at a mesh independent rate, when the prolongation norm is greater than one.
A Schwarz Preconditioner for a Hybridized Mixed Method
TLDR
A Schwarz preconditioner is provided for the linear equation for Lagrange multipliers arrived at by eliminating the flux as well as the primal variable for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods.
A projection-based error analysis of HDG methods
TLDR
A new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods, which renders the analysis of the projections of the discretization errors simple and concise.
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
TLDR
A unifying framework for hybridization of finite element methods for second order elliptic problems is introduced, thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.
The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms
TLDR
A theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms is provided and various numerical approximations of second-order elliptic boundary value problems are applied.
THE ANALYSIS OF SMOOTHERS FOR MULTIGRID ALGORITHMS
TLDR
The smoothing operators considered are based on subspace decomposition and include point, line, and block versions of Jacobi and Gauss-Seidel iteration as well as generalizations and it is shown that these smoothers will be effective in multigrid algorithms provided that the sub space decomposition satisfies two simple conditions.
HMG - Homogeneous multigrid for HDG
TLDR
A stable injection operator is constructed and optimal convergence of the method is proved under the assumption of elliptic regularity to introduce a homogeneous multigrid method for Poisson's equation on all levels.
Manycore Parallel Computing for a Hybridizable Discontinuous Galerkin Nested Multigrid Method
TLDR
This work presents a parallel computing strategy for a hybridizable discontinuous Galerkin (HDG) nested geometric multigrid (GMG) solver that combines a combination of coarse-grain and fine-grain components.
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