23/5/20051 ICCS congres, Atlanta, USA May 23, 2005 The Deflation Accelerated Schwarz Method for CFD C. Vuik Delft University of Technology J. Verkaik, B.D. Paarhuis, A. Twerda TNO Science and Industry

23/5/20052 Contents Problem description Schwarz domain decomposition Deflation GCR Krylov subspace acceleration Numerical experiments Conclusions

23/5/20053 Problem description CFD package TNO Science and Industry, The Netherlands simulation of glass melting furnaces incompressible Navier-Stokes equations, energy equation sophisticated physical models related to glass melting GTM-X:

23/5/20054 Problem description Incompressible Navier-Stokes equations: Discretisation: Finite Volume Method on “colocated” grid

23/5/20055 Problem description SIMPLE method: pressure- correction system ( )

23/5/20056 Schwarz domain decomposition Minimal overlap: Additive Schwarz:

23/5/20057 inaccurate solution to subdomain problems: 1 iteration SIP, SPTDMA or CG method complex geometries parallel computing local grid refinement at subdomain level solving different equations for different subdomains Schwarz domain decomposition GTM-X:

23/5/20058 Deflation: basic idea Solution: “remove” smallest eigenvalues that slow down the Schwarz method Problem: convergence Schwarz method deteriorates for increasing number of subdomains

23/5/20059 Deflation: deflation vectors +

23/5/ Property deflation method: systems with have to be solved by a direct method Deflation: Neumann problem singular Problem: pressure-correction matrix is singular: has eigenvector for eigenvalue 0 Solution: adjust non-singular 

23/5/ for general matrices (also singular) approximates in Krylov space such that is minimal Gram-Schmidt orthonormalisation for search directions optimisation of work and memory usage of Gram-Schmidt: restarting and truncating Additive Schwarz: Property: slow convergence Krylov acceleration GCR Krylov acceleration GCR Krylov method: Objective: efficient solution to

23/5/ Numerical experiments

23/5/ Numerical experiments Buoyancy-driven cavity flow

23/5/ Numerical experiments Buoyancy-driven cavity flow: inner iterations

23/5/ Numerical experiments Buoyancy-driven cavity flow: outer iterations without deflation

23/5/ Buoyancy-driven cavity flow: outer iterations with deflation Numerical experiments

23/5/ Buoyancy-driven cavity flow: outer iterations varying inner iterations Numerical experiments

23/5/ Numerical experiments Glass tank model

23/5/ Numerical experiments Glass tank model: inner iterations

23/5/ Numerical experiments Glass tank model: outer iterations without deflation

23/5/ Numerical experiments Glass tank model: outer iterations with deflation

23/5/ Glass tank model: outer iterations varying inner iterations Numerical experiments

23/5/ Heat conductivity flow Numerical experiments Q=0 Wm -2 h=30 Wm -2 K -1 T=303K T=1773K K = 1.0 Wm -1 K -1 K = 0.01 Wm -1 K -1 K = 100 Wm -1 K -1

23/5/ Heat conductivity flow: inner iterations Numerical experiments

23/5/ using linear deflation vectors seems most efficient a large jump in the initial residual norm can be observed higher convergence rates are obtained and wall-clock time can be gained implementation in existing software packages can be done with relatively low effort deflation can be applied to a wide range of problems Conclusions