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Register a new userSubstructuring Preconditioners for an h-p Nitsche-type method
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We propose and study an iterative substructuring method for an h-p Nitsche-type discretization, following the original approach introduced in [Bramble, Pasciack, Schatz (Math Comp. 1986)] for conforming methods. We prove quasi-optimality with respect to the mesh size and the polynomial degree for the proposed preconditioner. Numerical experiments asses the performance of the preconditioner and verify the theory.
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In this paper, we propose a deep unfitted Nitsche method for computing elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy function. We present several numerical examples to show the efficiency and accuracy of the proposed method.
We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are non-interpolatory, which makes the enforcement of the essential boundary conditions a nontrivial matter. Our method resorts to Nitsches variational formulation to deal with this difficulty, which is consistent, and does not require significant extra computational costs. We prove the error estimate in the energy norm and illustrate the method on several representative problems posed in at most 100 dimension.
The Sinc-Nystr{o}m method in time is a high-order spectral method for solving evolutionary differential equations and it has wide applications in scientific computation. But in this method we have to solve all the time steps implicitly at one-shot, which may results in a large-scale nonsymmetric dense system that is expensive to solve. In this paper, we propose and analyze a parallel-in-time (PinT) preconditioner for solving such Sinc-Nystr{o}m systems, where both the parabolic and hyperbolic PDEs are investigated. Attributed to the special Toeplitz-like structure of the Sinc-Nystr{o}m systems, the proposed PinT preconditioner is indeed a low-rank perturbation of the system matrix and we show that the spectrum of the preconditioned system is highly clustered around one, especially when the time step size is refined. Such a clustered spectrum distribution matches very well with the numerically observed mesh-independent GMRES convergence rates in various examples. Several linear and nonlinear ODE and PDE examples are presented to illustrate the convergence performance of our proposed PinT preconditioners, where the achieved exponential order of accuracy are especially attractive to those applications in need of high accuracy.
In this corrigendum, we offer a correction to [J. Korean. Math. Soc., 54 (2017), pp. 461--477]. We construct a counterexample for the strengthened Cauchy--Schwarz inequality used in the original paper. In addition, we provide a new proof for Lemma 5 of the original paper, an estimate for the extremal eigenvalues of the standard unpreconditioned FETI-DP dual operator.
In this paper, several projection method based preconditioners for various incompressible flow models are studied. In particular, we are interested in the theoretical analysis of a pressure-correction projection method based preconditioner cite{griffith2009accurate}. For both the steady and unsteady Stokes problems, we will show that the preconditioned systems are well conditioned. Moreover, when the flow model degenerates to the mixed form of an elliptic operator, the preconditioned system is an identity no matter what type of boundary conditions are imposed; when the flow model degenerates to the steady Stokes problem, the multiplicities of the non-unitary eigenvalues of the preconditioned system are derived. These results demonstrate the effects of boundary treatments and are related to the stability of the staggered grid discretization. To further investigate the effectiveness of these projection method based preconditioners, numerical experiments are given to compare their performances. Generalizations of these preconditioners to other saddle point problems will also be discussed.
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