Optimized Schwarz and 2-lagrange Multiplier Methods for Multiscale Pdes

نویسندگان

  • HIEU NGUYEN
  • ROBERT SCHEICHL
چکیده

In this article, we formulate and analyze a two-level preconditioner for Optimized Schwarz and 2-Lagrange Multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of low-frequency modes of the subdomain Dirichlet-to-Neumann maps. Under a suitable change of basis, the preconditioner is a 2 × 2 block upper triangular matrix with the identity matrix in the upper-left block. We show that the spectrum of the preconditioned system is included in the disk having center z = 1/2 and radius r = 1/2 − , where 0 < < 1/2 is a parameter that we can choose. We further show that the GMRES algorithm applied to our heterogeneous system converges in O(1/ ) iterations (neglecting certain polylogarithmic terms). The number can be made arbitrarily large by automatically enriching the coarse space. Our theoretical results are confirmed by numerical experiments.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Condition Number Estimates for the Nonoverlapping Optimized Schwarz Method and the 2-Lagrange Multiplier Method for General Domains and Cross Points

The optimized Schwarz method and the closely related 2-Lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations. Although these methods are known to work well in special cases (e.g., when the domain is a square and the two subdomains are rectangles), the problem has never been systematically stated nor analyzed f...

متن کامل

Primal interface formulation for coupling multiple PDEs: A consistent derivation via the Variational Multiscale method

This paper presents a primal interface formulation that is derived in a systematic manner from a Lagrange multiplier method to provide a consistent framework to couple different partial differential equations (PDE) as well as to tie together nonconforming meshes. The derivation relies crucially on concepts from the Variational Multiscale (VMS) approach wherein an additive multiscale decompositi...

متن کامل

Sharp Condition Number Estimates for the Symmetric 2-Lagrange Multiplier Method

Domain decomposition methods are used to find the numerical solution of large boundary value problems in parallel. In optimized domain decomposition methods, one solves a Robin subproblem on each subdomain, where the Robin parameter a must be tuned (or optimized) for good performance. We show that the 2-Lagrange multiplier method can be analyzed using matrix analytical techniques and we produce...

متن کامل

A Schwarz Preconditioner for a Hybridized Mixed Method

In this paper, we provide a Schwarz preconditioner for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods. The preconditioner is for the linear equation for Lagrange multipliers arrived at by eliminating the flux as well as the primal variable. We also prove a condition number estimate for this equation when no preconditioner is used. Although preconditioners ...

متن کامل

Efficient Multiresolution Counterparts to Variational Methods for Surface Reconstruction

Variational methods have been employed with considerable success in computer vision, particularly for surface reconstruction problems. Formulations of this type require the solution of computationally complex Euler–Lagrange partial differential equations (PDEs) to obtain the desired reconstructions. Further, the calculation of reconstruction error covariances for such approaches are usually neg...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2014