Numerical Linear Algebra: Numerical solutions to Stokes Equation

This repository implements the final project for the course Numerical Linear Algebra in Autumn 2018. Submitted version.

Description of codes

The main codes of my implementations are stored under 'Matlab_Code'. Certain auxiliary programs which help us build the matrix for the problem are catogorized into components. The notations are consistent with my report.

The main program is VCycle.m . It implements V-Cycle multigrid method to solve the saddle point problem, which is the discretization of the Stokes Equation.

The smoothers contain:

• DGS-p (in Long Chen's paper): DGS.m.
• DGS-s (in the notes): DGS_seq.m and DGS_seqM.m. Their inner loop programs are DGS_InnerC2.c and DGS_InnerM.m . The former is implemented in C and the latter is implemenented in Matlab. DGS_InnerC.c is the history version of DGS_InnerC2.c.
• Uzawa: Uzawa.m. We simply use \ in Matlab to solve the subproblem.
• Inexact Uzawa: Inexact_Uzawa.m. We call CG_solver.m to solve the subproblem.

We also implement Inexact Uzawa based on V-Cycle multi-grid. The main program is ieuzawaVC.m. We use V-Cycle multi-grid method to solve the subproblem. This is implemented in VCycle_Inner.m. The smoother for V-Cycle is Gauss-Seidel Iteration, which is implemented in GS.m.

In Appendix, we propose a modified V-Cycle multi-grid method based on DGS. The main program is VCycle_mod.m. The smoothers contain:

• DGS-p(in Long Chen's paper): DGS_mod.m.

• DGS-s (in the notes): DGS_seqM_mod.m. Its innner loop program is DGS_InnerM_mod.m

How to run the codes

To run the codes, first, please use MEX in Matlab to compile DGS_innerC2.c in the following way:

mex DGS_innerC2.c

To recover the results for DGS-s and DGS-p, please run:

• $N=64, 128, 256$: Test_DGS1.m (DGS-s), Test_DGS2.m (DGS-p);
• $N=512, 1024, 2048$: Test_DGS3.m (DGS-s), Test_DGS4.m (DGS-p).

To recover the results for Uzawa, please run:

• $N = 64, 128$ : Test_uzawa.m

• $N = 256, 512$ : Test_uzawa2.m

• $N = 1024, 2048$ : Test_uzawa3.m

To recover the results for Inexact Uzawa, please run:

• $N = 64, 128$ : Test_ieuzawa.m

• $N = 256, 512$ : Test_ieuzawa2.m

• $N = 1024, 2048$ : Test_ieuzawa3.m

To recover the results for Inexact Uzawa based on V-Cycle, please run:

• $N = 64, 128$ : Test_ieuzawaVC.m

• $N = 256, 512$ : Test_ieuzawaVC2.m

• $N = 1024, 2048$ : Test_ieuzawaVC3.m

Catogory

Our Matlab programs are generally catogorized as follows:

bnd

'comp_*.m' generates the vector to describe the Neurman boundary condition

comp

'comp_*.m' generates matrix which is component in the formulation of saddle point problem.

spp

'comp_*.m' generates matrix in the saddle point problem

test

'test_*.m' implements the numerical experiements in the report

true

'true_*.m' returns the true value of the function