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Register a new userManycore parallel computing for a hybridizable discontinuous Galerkin nested multigrid method
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Added by
Maurice S. Fabien
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We present a parallel computing strategy for a hybridizable discontinuous Galerkin (HDG) nested geometric multigrid (GMG) solver. Parallel GMG solvers require a combination of coarse-grain and fine-grain parallelism to improve time to solution performance. In this work we focus on fine-grain parallelism. We use Intels second generation Xeon Phi (Knights Landing) many-core processor. The GMG method achieves ideal convergence rates of $0.2$ or less, for high polynomial orders. A matrix free (assembly free) technique is exploited to save considerable memory usage and increase arithmetic intensity. HDG enables static condensation, and due to the discontinuous nature of the discretization, we developed a matrix vector multiply routine that does not require any costly synchronizations or barriers. Our algorithm is able to attain 80% of peak bandwidth performance for higher order polynomials. This is possible due to the data locality inherent in the HDG method. Very high performance is realized for high order schemes, due to good arithmetic intensity, which declines as the order is reduced.
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In this paper, we develop a nested hybridizable discontinuous Galerkin (HDG) method to numerically solve the Maxwells equations coupled with the hydrodynamic model for the conduction-band electrons in metals. By means of a static condensation to eliminate the degrees of freedom of the approximate solution defined in the elements, the HDG method yields a linear system in terms of the degrees of freedom of the approximate trace defined on the element boundaries. Furthermore, we propose to reorder these degrees of freedom so that the linear system accommodates a second static condensation to eliminate a large portion of the degrees of freedom of the approximate trace, thereby yielding a much smaller linear system. For the particular metallic structures considered in this paper, the resulting linear system obtained by means of nested static condensations is a block tridiagonal system, which can be solved efficiently. We apply the nested HDG method to compute the second harmonic generation (SHG) on a triangular coaxial periodic nanogap structure. This nonlinear optics phenomenon features rapid field variations and extreme boundary-layer structures that span multiple length scales. Numerical results show that the ability to identify structures which exhibit resonances at $omega$ and $2omega$ is paramount to excite the second harmonic response.
We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced, allowing to derive alternative guaranteed bounds from nearly-arbitrary H(div;{Omega}) flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements.
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