update project Stokes MAC

Author: lyc102

docs/_projects/projectStokesMAC.md

@@ -6,60 +6,69 @@ sidebar:
 ---
 
 
-The purpose of this project is to implement the simple and popular MAC scheme for solving Stokes equations in two dimensions. 
+The purpose of this project is to implement the MAC scheme for solving Stokes equations in two dimensions. 
 
 Reference
-* [Progamming of MAC for Stokes equations](http://math.uci.edu/~chenlong/226/MACcode.pdf)
+* [Progamming of MAC for Stokes Equations](http://math.uci.edu/~chenlong/226/MACcode.pdf)
+* [Progamming of Finite Difference Methods](http://math.uci.edu/~chenlong/226/FDMcode.pdf)
 
-## Test Examples
 
-* Analytic solution. We use a simple model of colliding flow with
-analytic solutions to test the code. The domain is $[-1,1]^2$  Compute the data `f` and Dirichlet boundary condition `g_D` for the analytic solution:
 
-$$u = 20xy^3; \quad v = 5x^4 - 5y^4; \quad p = 60x^2y - 20y^3 + {\rm constant}.$$
+## Step 1: Distributive Gauss-Seidel Smoothing (DGS)
 
-* Driven cavity problem. The domain is [-1,1]^2. Stokes equation with zero Dirichlet boundary condition except on the top:
-
-$$ \{ y=1; -1 <= x <= 1 | u = 1, v = 0 \}.$$
-
-## Step 1: Gauss-Seidel smoothing of velocity
+1. Gauss-Seidel smoothing of velocity to update `u`;
+2. form the residual for the continuity equation `rp = g-Bu`;
+3. solve the Poisson equation for pressure `Ap*ep = rp` by G-S;
+4. distribute the correction to velocity by `u = u + B'*ep`;
+5. update the pressure by `p = p - Ap*ep`.
 
-Given a pressure approximation, relax the momentum equation to update velocity. See [Project: Multigrid Methods](projectMG.md) on the matrix free implemenation of G-S relaxation.
+Every step can be implemented in a matrix-free version.
 
-Note that the boundary or near boundary dof should be updated diffeently. The stencil should be changed according to different boundary conditions.
+Then use DGS as an iterative method to solve the Stokes equation for a fixed level. As the level increases, the iteration steps will increase to achieve the same tolerance, e.g. the relative residual norm is less than $10^{-3}$​. 
 
-# Step 2: Distributive Gauss-Seidel Smoothing (DGS)
+To debug your code, let the exact solution be zero. Start from a random initial guess. The DGS iteration will push all unknowns to zero. 
 
-1. Gauss-Seidel smoothing of velocity to update `u`;
-* form the residual for the continuity equation `rp = g-Bu`;
-* solve the Poisson equation for pressure `Ap*ep = rp` by G-S;
-* distribute the correction to velocity by `u = u + B'*ep`;
-* update the pressure by `p = p - Ap*ep`.
 
-Every step can be implemented in a matrix-free version; see  [Progamming of MAC for Stokes equations](http://math.uci.edu/~chenlong/226/MACcode.pdf).
 
-Use DGS as an iterative method to solve the Stokes equation. The iteration steps could be very large but will converges.
-
-# Step 3: Two level method
+## Step 2: Two Level Method
 
 The two level method is
 
-1. presmoothing by DGS
-* form residuals for momentum and continunity equations
-* restrict the residual to the coarse grid
-* iterate DGS in the coarse grid till converge
-* prolongate the correction to the fine grid
-* postsmoothing by DGS
+1. Presmoothing by DGS
+2. Form residuals for momentum and continunity equations
+3. Restrict the residual to the coarse grid
+4. Use DGS in the coarse grid till converge (Step 1)
+5. Prolongate the correction to the fine grid
+6. Postsmoothing by DGS
 
-Note: the index map between coarse and fine grids are slightly different for `u,v,p`; see  [Progamming of MAC for Stokes equations](http://math.uci.edu/~chenlong/226/MACcode.pdf).
+Note: the index map between coarse and fine grids are slightly different for `u,v,p`.
 
 Test your two level methods for different levels. It should convergence in less than 20 steps and indepedent of the number of levels.
 
-## Step 4: Vcycle multigrid method
 
-Recrusively apply the two-level method to the coarse grid problem
-in the previous step to get a V-cycle method.
 
-* Test the convergence of Vcycle method. Record the iteration steps needed to push the relative residual smaller than a tolerance.
+## Step 3: Vcycle Multigrid Method
+
+Recrusively apply the two-level method to the coarse grid problem in the previous step to get a V-cycle method.
+
+* Test the convergence of Vcycle method. Record the iteration steps needed to push the relative residual smaller than a tolerance say $10^{-3}$.
 
 * Compute the error between the computed approximation to the exact solution and show the convergence rate in terms of mesh size `h`. 
+
+
+
+## Step 4: Test Examples
+
+1. **Analytic solution.** We use a simple model of colliding flow with analytic solutions to test the code. The domain is $[-1,1]^2$​  Compute the data `f` and Dirichlet boundary condition `g_D` for the analytic solution:
+   $$
+   u = 20xy^3; \quad v = 5x^4 - 5y^4; \quad p = 60x^2y - 20y^3 + {\rm constant}.
+   $$
+   Choose the constant s.t. the integral of $p$ over the domain is zero.
+
+2. **Driven cavity problem.** The domain is $[-1,1]^2$​ . Stokes equation with $f=0,g=0$ and zero Dirichlet boundary condition except on the top:
+
+$$
+\{ y=1; -1 \leq x \leq 1 \mid u = 1, v = 0 \}.
+$$
+
+## 

tutorial/mgS.m

@@ -41,7 +41,7 @@
 
 %% Options
 % Assign default values to unspecified parameters
-if ~exist('option','var'), 
+if ~exist('option','var')
     option = []; 
 end
 option = mgoptions(option,Ndof);    % parameters