--- a/+scheme/Heat2dVariable.m	Tue Dec 04 14:54:28 2018 -0800
+++ b/+scheme/Heat2dVariable.m	Tue Dec 04 15:24:36 2018 -0800
@@ -1,9 +1,9 @@
 classdef Heat2dVariable < scheme.Scheme
 
 % Discretizes the Laplacian with variable coefficent,
-% In the Heat equation way (i.e., the discretization matrix is not necessarily
+% In the Heat equation way (i.e., the discretization matrix is not necessarily 
 % symmetric)
-% u_t = div * (kappa * grad u )
+% u_t = div * (kappa * grad u ) 
 % opSet should be cell array of opSets, one per dimension. This
 % is useful if we have periodic BC in one direction.
 
@@ -28,7 +28,8 @@
         H, Hi % Inner products
         e_l, e_r
         d1_l, d1_r % Normal derivatives at the boundary
-
+        alpha % Vector of borrowing constants
+        
         H_boundary % Boundary inner products
 
     end
@@ -144,6 +145,7 @@
             obj.order = order;
             obj.grid = g;
             obj.dim = dim;
+            obj.alpha = [ops{1}.borrowing.M.d1, ops{2}.borrowing.M.d1];
 
         end
 
@@ -155,12 +157,13 @@
         %       data                is a function returning the data that should be applied at the boundary.
         %       neighbour_scheme    is an instance of Scheme that should be interfaced to.
         %       neighbour_boundary  is a string specifying which boundary to interface to.
-        function [closure, penalty] = boundary_condition(obj, boundary, type, parameter)
+        function [closure, penalty] = boundary_condition(obj, boundary, type, symmetric, tuning)
             default_arg('type','Neumann');
-            default_arg('parameter', []);
+            default_arg('symmetric', false);
+            default_arg('tuning',1.2);
 
             % j is the coordinate direction of the boundary
-            % nj: outward unit normal component.
+            % nj: outward unit normal component. 
             % nj = -1 for west, south, bottom boundaries
             % nj = 1  for east, north, top boundaries
             [j, nj] = obj.get_boundary_number(boundary);
@@ -176,19 +179,30 @@
             Hi = obj.Hi;
             H_gamma = obj.H_boundary{j};
             KAPPA = obj.KAPPA;
-            kappa_gamma = e{j}'*KAPPA*e{j};
+            kappa_gamma = e{j}'*KAPPA*e{j}; 
+            h = obj.h(j);
+            alpha = h*obj.alpha(j);
 
             switch type
 
             % Dirichlet boundary condition
             case {'D','d','dirichlet','Dirichlet'}
-                    closure = -nj*Hi*d{j}*kappa_gamma*H_gamma*(e{j}' );
+
+                if ~symmetric
+                    closure = -nj*Hi*d{j}*kappa_gamma*H_gamma*(e{j}' ); 
                     penalty =  nj*Hi*d{j}*kappa_gamma*H_gamma;
+                else
+                    closure = nj*Hi*d{j}*kappa_gamma*H_gamma*(e{j}' )...
+                              -tuning*2/alpha*Hi*e{j}*kappa_gamma*H_gamma*(e{j}' ) ; 
+                    penalty =  -nj*Hi*d{j}*kappa_gamma*H_gamma ...
+                              +tuning*2/alpha*Hi*e{j}*kappa_gamma*H_gamma;
+                end
 
             % Free boundary condition
             case {'N','n','neumann','Neumann'}
-                    closure = -nj*Hi*e{j}*kappa_gamma*H_gamma*(d{j}' );
-                    penalty =  nj*Hi*e{j}*kappa_gamma*H_gamma;
+                    closure = -nj*Hi*e{j}*kappa_gamma*H_gamma*(d{j}' ); 
+                    penalty =  Hi*e{j}*kappa_gamma*H_gamma; 
+                    % penalty is for normal derivative and not for derivative, hence the sign.
 
             % Unknown boundary condition
             otherwise
@@ -196,7 +210,7 @@
             end
         end
 
-        function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary,opts)
+        function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
             % u denotes the solution in the own domain
             % v denotes the solution in the neighbour domain
             error('Interface not implemented');