Schwarz alternating methods for anisotropic problems with prolate spheroid boundaries

Dai, Zhenlong; Du, Qikui; Liu, Baoqing

doi:10.1186/s40064-016-3063-y

Cited by 2 publications

(2 citation statements)

References 20 publications

(25 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Optimized Schwarz methods are a modern class of Schwarz methods which use instead of the classical Dirichlet transmission conditions at the interfaces more effective transmission conditions, which can take the physics of the problem at hand into account, see [18,19] and references therein. This property is especially important for anisotropic diffusion problems, which behave very differently at interfaces depending on the orientation of the diffusion, see for example [24], [14,Section 5], and the very recent reference [35]; for classical Schwarz methods applied to anisotropic diffusion, see [36,8,11], and for a specific earlier two level preconditioner [32]. Similarly when discretizing anisotropic diffusion problems, the numerical scheme must be suitable for high anisotropy, and discrete duality finite volume (DDFV) methods have this property, even in the case of discontinuous anisotropic diffusion, see [27,5,26,6,9,15,2], and in particular [16,Part II] which is dedicated especially to anisotropic diffusion.…”

Section: Introductionmentioning

confidence: 99%

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Gander

Halpern

Hubert

et al. 2020

Moroccan Journal of Pure and Applied Analysis

View full text Add to dashboard Cite

We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.

show abstract

Section: Introductionmentioning

confidence: 99%

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Gander

Halpern

Hubert

et al. 2020

Moroccan Journal of Pure and Applied Analysis

View full text Add to dashboard Cite

show abstract

“…The first approach is to apply the traditional iterative Schwarz algorithms first developed for elliptic problems [33,34,35,36,37,38] to the elliptic equations which arise upon semi-discretizing the time-dependent PDE in time (see [39,40,41,42,43,44,45]). The second approach is to split the whole space-time domain into overlapping or non-overlapping space-time subdomains in a Schwarz waveform relaxation framework [34,46,47,48]. The third approach is noniterative domain decomposition which is used to further reduce the computational cost [49,50,51,52,53,54].…”

Section: Introductionmentioning

confidence: 99%

Domain Decomposition Parabolic Monge-Ampère Approach for Fast Generation of Adaptive Moving Meshes

Sulman

Nguyen

Haynes

et al. 2020

Preprint

View full text Add to dashboard Cite

A fast method is presented for adaptive moving mesh generation in multidimensions using a domain decomposition parabolic Monge-Ampère approach. The domain decomposition procedure employed here is non-iterative and involves splitting the computational domain into overlapping subdomains. An adaptive mesh on each subdomain is then computed as the image of the solution of the L 2 optimal mass transfer problem using a parabolic Monge-Ampère method. The domain decomposition approach allows straightforward implementation for the parallel computation of adaptive meshes which helps to reduce computational time significantly. Results are presented to show the numerical convergence of the domain decomposition solution to the single domain solution. Several numerical experiments are given to demonstrate the performance and efficiency of the proposed method. The numerical results indicate that the domain decomposition parabolic Monge-Ampère method is more efficient than the standard implementation of the parabolic Monge-Ampère method on the whole domain, in particular when computing adaptive meshes in three spatial dimensions.

show abstract

Schwarz alternating methods for anisotropic problems with prolate spheroid boundaries

Cited by 2 publications

References 20 publications

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Domain Decomposition Parabolic Monge-Ampère Approach for Fast Generation of Adaptive Moving Meshes

Contact Info

Product

Resources

About