XT_DG for the 1D wave equation, cartesian mesh only Domain decomposition methods: RAS, GMRES with RAS preconditioner, RAS PIPELINE

from A space-time discontinuous Galerkin method for the wave equation Authors: Francesco Songia, Enrico Zardi mailto: [email protected], [email protected] Advisors: Ilario Mazzieri, Gabriele Ciaramella Project in Numerical Analysis for Partial Differential Equations Academic year 2021-22

In this file:

• code main structure
• 'tests' folder description
• result structures descriptions

For the DG discretization we have used as a base structure a given library for solving acoustic poroelastic problems with polygonal meshes. We have kept only the acoustic structure but then we have modified all the matrices computation since we exploit a space time strategy. Those was provided by our advisors. We have created from scratch the domain decomposition library (DomainDec) with all the methods.

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Code main structure

main.m DG discretization (XT_DG_run.m)

domain decomposition (run_domainDec_withA.m)

residual evolution in dd methods

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'tests' folder

In the folder 'tests' we stored the results of domain decomposition methods (if saveinfo = 1 in main.m). In this folder it is created a new folder named with the creation time, the mesh used (es mesh_0105_10020 stands for a [0,1]x[0,5] domain (space x time) with 100 and 20 finite elements for space and time respectively), the formulation used {IP,IPH} and the DG stability coefficient mu.

k = number of domain decomposition choice exploited in the test Inside this new folder there are:

• k .png images with the decompsed domain
• k .mat files named as the mesh (backup of a single decomposition choice). Here there is the backup struct with all the test information
• test_result (.mat and .xls) with a compact description of all results

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Result structures description

test_results

tests_result=[info,i,dd,result_ras,result_gmres,result_pipe] (is a vector) info: [X,T,NX,NT,prob,mu,alpha,form,it_max,it_max_pipe,tol,tol_sx,it_wait] i: index of test with i=1,..,k, k = how many decomposition choice dd: [nsub_x,nsub_t,m,n,ot,ox] result_ras: [it_ras, time_ras, relres2P_vec_ras(end),relresinfP_vec_ras(end), relresinf_vec_ras(end),solved_dom_ras]
result_gmres: [it_gmres,time_gmres,relres2P_vec_gmres(end),solved_dom_gmres]
result_pipe: [it_pipe,time_pipe,resinf_vec_pipe(end),solved_dom_pipe]

X,T: domain [0,X]x[0,T] NX, NT: finite elements prob: number of input data problem (InputData\DataTest.m) mu, alpha: DG stability coeff, mu = alpha * fem_degree/h, h = X/NX form: IP (0) or IPH (1) formulation it_max: max number of iterations for RAS, GMRES it_max_pipe: max number of iterations for RAS PIPELINE tol: tolerance for convergence tol_sx: only for RAS PIPELINE. Tolerance to reach for move the left edge of the subdomains window it_wait: only for RAS PIPELINE. Wait it_wait iterations before update the right edge
of the subdomains window i: index of test with i=1,..,k, k = how many decomposition choice nsub_x,nsub_t: number of subdomains m, n: each subdomain is composed by nm finite elements (spacetime) ot, ox: number of finite elements that form the overlap. They are an average if
there isn't a perfect decomposition result_***: contains iterations used, time, final residuals and solved subdomains. relres2P stands for relative residual L2 norm, P stands for preconditioned residual.

backup

 info: the same as above,
 dd: the same as above,
 perf: performance of the methods. It contains the same result as above but  
       there are all the residual vectors (all the history).

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