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Register a new userA GenEO Domain Decomposition method for Saddle Point problems
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Added by
Frederic Nataf
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom.
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This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential equations (PDE). Using DNN to solve PDE is a physics-informed learning problem with the objective involving two terms, domain term and boundary term, which respectively make the desired solution satisfy the PDE and corresponding boundary conditions. DeepDDM will exchange the subproblem information across the interface in DDM by adjusting the boundary term for solving each subproblem by DNN. Benefiting from the simple implementation and mesh-free strategy of using DNN for PDE, DeepDDM will simplify the implementation of DDM and make DDM more flexible for complex PDE, e.g., those with complex interfaces in the computational domain. This paper will firstly investigate the performance of using DeepDDM for elliptic problems, including a model problem and an interface problem. The numerical examples demonstrate that DeepDDM exhibits behaviors consistent with conventional DDM: the number of iterations by DeepDDM is independent of network architecture and decreases with increasing overlapping size. The performance of DeepDDM on elliptic problems will encourage us to further investigate its performance for other kinds of PDE and may provide new insights for improving the PDE solver by deep learning.
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In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in 3D effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness and the stability estimate of the PML problem are first proved based on the Laplace transform and the energy method. The exponential convergence of the PML method is then established in terms of the thickness of the layer and the PML absorbing parameter. As far as we know, this is the first convergence result for the time-domain PML method for the three-dimensional Maxwell equations. Our proof is mainly based on the stability estimates of solutions of the truncated PML problem and the exponential decay estimates of the stretched dyadic Greens function for the Maxwell equations in the free space.
In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the $L^2$-norm and $L^{infty}$-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (SIAM J. Numer. Anal. 58(3) (2020), 1918-1940).
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