--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/LaplaceCurvilinearNewCorner.m	Thu Jan 17 18:57:54 2019 -0800
@@ -0,0 +1,579 @@
+classdef LaplaceCurvilinearNewCorner < scheme.Scheme
+    properties
+        m % Number of points in each direction, possibly a vector
+        h % Grid spacing
+        dim % Number of spatial dimensions
+
+        grid
+
+        order % Order of accuracy for the approximation
+
+        a,b % Parameters of the operator
+
+
+        % Inner products and operators for physical coordinates
+        D % Laplace operator
+        H, Hi % Inner product
+        e_w, e_e, e_s, e_n
+        d_w, d_e, d_s, d_n % Normal derivatives at the boundary
+        H_w, H_e, H_s, H_n % Boundary inner products
+        Dx, Dy % Physical derivatives
+        M % Gradient inner product
+
+        % Metric coefficients
+        J, Ji
+        a11, a12, a22
+        K
+        x_u
+        x_v
+        y_u
+        y_v
+        s_w, s_e, s_s, s_n % Boundary integral scale factors
+
+        % Inner product and operators for logical coordinates
+        H_u, H_v % Norms in the x and y directions
+        Hi_u, Hi_v
+        Hu,Hv % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
+        Hiu, Hiv
+        du_w, dv_w
+        du_e, dv_e
+        du_s, dv_s
+        du_n, dv_n
+
+        % Borrowing constants
+        theta_M_u, theta_M_v
+        theta_R_u, theta_R_v
+        theta_H_u, theta_H_v
+
+    end
+
+    methods
+        % Implements  a*div(b*grad(u)) as a SBP scheme
+        % TODO: Implement proper H, it should be the real physical quadrature, the logic quadrature may be but in a separate variable (H_logic?)
+
+        function obj = LaplaceCurvilinearNewCorner(g, order, a, b, opSet)
+            default_arg('opSet',@sbp.D2Variable);
+            default_arg('a', 1);
+            default_arg('b', 1);
+
+            if b ~=1
+                error('Not implemented yet')
+            end
+
+            % assert(isa(g, 'grid.Curvilinear'))
+            if isa(a, 'function_handle')
+                a = grid.evalOn(g, a);
+                a = spdiag(a);
+            end
+
+            dim = 2;
+            m = g.size();
+            m_u = m(1);
+            m_v = m(2);
+            m_tot = g.N();
+
+            h = g.scaling();
+            h_u = h(1);
+            h_v = h(2);
+
+
+            % 1D operators
+            ops_u = opSet(m_u, {0, 1}, order);
+            ops_v = opSet(m_v, {0, 1}, order);
+
+            I_u = speye(m_u);
+            I_v = speye(m_v);
+
+            D1_u = ops_u.D1;
+            D2_u = ops_u.D2;
+            H_u =  ops_u.H;
+            Hi_u = ops_u.HI;
+            e_l_u = ops_u.e_l;
+            e_r_u = ops_u.e_r;
+            d1_l_u = ops_u.d1_l;
+            d1_r_u = ops_u.d1_r;
+
+            D1_v = ops_v.D1;
+            D2_v = ops_v.D2;
+            H_v =  ops_v.H;
+            Hi_v = ops_v.HI;
+            e_l_v = ops_v.e_l;
+            e_r_v = ops_v.e_r;
+            d1_l_v = ops_v.d1_l;
+            d1_r_v = ops_v.d1_r;
+
+
+            % Logical operators
+            Du = kr(D1_u,I_v);
+            Dv = kr(I_u,D1_v);
+            obj.Hu  = kr(H_u,I_v);
+            obj.Hv  = kr(I_u,H_v);
+            obj.Hiu = kr(Hi_u,I_v);
+            obj.Hiv = kr(I_u,Hi_v);
+
+            e_w  = kr(e_l_u,I_v);
+            e_e  = kr(e_r_u,I_v);
+            e_s  = kr(I_u,e_l_v);
+            e_n  = kr(I_u,e_r_v);
+            obj.du_w = kr(d1_l_u,I_v);
+            obj.dv_w = (e_w'*Dv)';
+            obj.du_e = kr(d1_r_u,I_v);
+            obj.dv_e = (e_e'*Dv)';
+            obj.du_s = (e_s'*Du)';
+            obj.dv_s = kr(I_u,d1_l_v);
+            obj.du_n = (e_n'*Du)';
+            obj.dv_n = kr(I_u,d1_r_v);
+
+
+            % Metric coefficients
+            coords = g.points();
+            x = coords(:,1);
+            y = coords(:,2);
+
+            x_u = Du*x;
+            x_v = Dv*x;
+            y_u = Du*y;
+            y_v = Dv*y;
+
+            J = x_u.*y_v - x_v.*y_u;
+            a11 =  1./J .* (x_v.^2  + y_v.^2);
+            a12 = -1./J .* (x_u.*x_v + y_u.*y_v);
+            a22 =  1./J .* (x_u.^2  + y_u.^2);
+            % lambda = 1/2 * (a11 + a22 - sqrt((a11-a22).^2 + 4*a12.^2));
+
+            K = cell(dim, dim);
+            K{1,1} = spdiag(y_v./J);
+            K{1,2} = spdiag(-y_u./J);
+            K{2,1} = spdiag(-x_v./J);
+            K{2,2} = spdiag(x_u./J);
+            obj.K = K;
+
+            obj.x_u = x_u;
+            obj.x_v = x_v;
+            obj.y_u = y_u;
+            obj.y_v = y_v;
+
+            % Assemble full operators
+            L_12 = spdiag(a12);
+            Duv = Du*L_12*Dv;
+            Dvu = Dv*L_12*Du;
+
+            Duu = sparse(m_tot);
+            Dvv = sparse(m_tot);
+            ind = grid.funcToMatrix(g, 1:m_tot);
+
+            for i = 1:m_v
+                D = D2_u(a11(ind(:,i)));
+                p = ind(:,i);
+                Duu(p,p) = D;
+            end
+
+            for i = 1:m_u
+                D = D2_v(a22(ind(i,:)));
+                p = ind(i,:);
+                Dvv(p,p) = D;
+            end
+
+
+            % Physical operators
+            obj.J = spdiag(J);
+            obj.Ji = spdiag(1./J);
+
+            obj.D = obj.Ji*a*(Duu + Duv + Dvu + Dvv);
+            obj.H = obj.J*kr(H_u,H_v);
+            obj.Hi = obj.Ji*kr(Hi_u,Hi_v);
+
+            obj.e_w = e_w;
+            obj.e_e = e_e;
+            obj.e_s = e_s;
+            obj.e_n = e_n;
+
+            %% normal derivatives
+            I_w = ind(1,:);
+            I_e = ind(end,:);
+            I_s = ind(:,1);
+            I_n = ind(:,end);
+
+            a11_w = spdiag(a11(I_w));
+            a12_w = spdiag(a12(I_w));
+            a11_e = spdiag(a11(I_e));
+            a12_e = spdiag(a12(I_e));
+            a22_s = spdiag(a22(I_s));
+            a12_s = spdiag(a12(I_s));
+            a22_n = spdiag(a22(I_n));
+            a12_n = spdiag(a12(I_n));
+
+            s_w = sqrt((e_w'*x_v).^2 + (e_w'*y_v).^2);
+            s_e = sqrt((e_e'*x_v).^2 + (e_e'*y_v).^2);
+            s_s = sqrt((e_s'*x_u).^2 + (e_s'*y_u).^2);
+            s_n = sqrt((e_n'*x_u).^2 + (e_n'*y_u).^2);
+
+            obj.d_w = -1*(spdiag(1./s_w)*(a11_w*obj.du_w' + a12_w*obj.dv_w'))';
+            obj.d_e =    (spdiag(1./s_e)*(a11_e*obj.du_e' + a12_e*obj.dv_e'))';
+            obj.d_s = -1*(spdiag(1./s_s)*(a22_s*obj.dv_s' + a12_s*obj.du_s'))';
+            obj.d_n =    (spdiag(1./s_n)*(a22_n*obj.dv_n' + a12_n*obj.du_n'))';
+
+            obj.Dx = spdiag( y_v./J)*Du + spdiag(-y_u./J)*Dv;
+            obj.Dy = spdiag(-x_v./J)*Du + spdiag( x_u./J)*Dv;
+
+            %% Boundary inner products
+            obj.H_w = H_v*spdiag(s_w);
+            obj.H_e = H_v*spdiag(s_e);
+            obj.H_s = H_u*spdiag(s_s);
+            obj.H_n = H_u*spdiag(s_n);
+
+            % Misc.
+            obj.m = m;
+            obj.h = [h_u h_v];
+            obj.order = order;
+            obj.grid = g;
+            obj.dim = dim;
+
+            obj.a = a;
+            obj.b = b;
+            obj.a11 = a11;
+            obj.a12 = a12;
+            obj.a22 = a22;
+            % obj.lambda = lambda;
+            obj.s_w = spdiag(s_w);
+            obj.s_e = spdiag(s_e);
+            obj.s_s = spdiag(s_s);
+            obj.s_n = spdiag(s_n);
+
+            obj.theta_M_u = h_u*ops_u.borrowing.M.d1;
+            obj.theta_M_v = h_v*ops_v.borrowing.M.d1;
+
+            obj.theta_R_u = h_u*ops_u.borrowing.R.delta_D;
+            obj.theta_R_v = h_v*ops_v.borrowing.R.delta_D;
+
+            obj.theta_H_u = h_u*ops_u.borrowing.H11;
+            obj.theta_H_v = h_v*ops_v.borrowing.H11;
+        end
+
+
+        % Closure functions return the opertors applied to the own doamin to close the boundary
+        % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin.
+        %       boundary            is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
+        %       type                is a string specifying the type of boundary condition if there are several.
+        %       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)
+            default_arg('type','neumann');
+            default_arg('parameter', []);
+
+            [e, d] = obj.getBoundaryOperator({'e', 'd'}, boundary);
+            H_b = obj.getBoundaryQuadrature(boundary);
+            s_b = obj.getBoundaryScaling(boundary);
+            [th_H, th_M, th_R] = obj.getBoundaryBorrowing(boundary);
+            m = obj.getBoundaryNumber(boundary);
+
+            K = obj.K;
+            J = obj.J;
+            Hi = obj.Hi;
+            a_b = e'*obj.a*e;
+
+            switch type
+                % Dirichlet boundary condition
+                case {'D','d','dirichlet'}
+                    tuning = 1.0;
+
+                    sigma = 0;
+                    for i = 1:obj.dim
+                        sigma = sigma + e'*J*K{i,m}*K{i,m}*e;
+                    end
+                    sigma = sigma/s_b;
+                    % tau = tuning*(1/th_R + obj.dim/th_H)*sigma;
+
+                    tau_R = 1/th_R*sigma;
+
+                    tau_H = 1/th_H*sigma;
+                    tau_H(1,1) = obj.dim*tau_H(1,1);
+                    tau_H(end,end) = obj.dim*tau_H(end,end);
+
+                    tau = tuning*(tau_R + tau_H);
+
+                    closure = Hi*d*a_b*H_b*e' ...
+                             -Hi*e*tau*H_b*e';
+
+                    penalty = -Hi*d*a_b*H_b ...
+                              +Hi*e*tau*H_b;
+
+
+                % Neumann boundary condition
+                case {'N','n','neumann'}
+                    tau1 = -1;
+                    tau2 = 0;
+                    tau = (tau1*e + tau2*d)*H_b;
+
+                    closure =  obj.a*Hi*tau*d';
+                    penalty = -obj.a*Hi*tau;
+
+
+                % Unknown, boundary condition
+                otherwise
+                    error('No such boundary condition: type = %s',type);
+            end
+        end
+
+        % type     Struct that specifies the interface coupling.
+        %          Fields:
+        %          -- tuning:           penalty strength, defaults to 1.2
+        %          -- interpolation:    type of interpolation, default 'none'
+        function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary,type)
+
+            error('Not implemented')
+
+            defaultType.tuning = 1.2;
+            defaultType.interpolation = 'none';
+            default_struct('type', defaultType);
+
+            switch type.interpolation
+            case {'none', ''}
+                [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary,type);
+            case {'op','OP'}
+                [closure, penalty] = interfaceNonConforming(obj,boundary,neighbour_scheme,neighbour_boundary,type);
+            otherwise
+                error('Unknown type of interpolation: %s ', type.interpolation);
+            end
+        end
+
+        function [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary,type)
+            tuning = type.tuning;
+
+            % u denotes the solution in the own domain
+            % v denotes the solution in the neighbour domain
+            [e_u, d_u] = obj.getBoundaryOperator({'e', 'd'}, boundary);
+            H_b_u = obj.getBoundaryQuadrature(boundary);
+            I_u = obj.getBoundaryIndices(boundary);
+            gamm_u = obj.getBoundaryBorrowing(boundary);
+
+            [e_v, d_v] = neighbour_scheme.getBoundaryOperator({'e', 'd'}, neighbour_boundary);
+            H_b_v = neighbour_scheme.getBoundaryQuadrature(neighbour_boundary);
+            I_v = neighbour_scheme.getBoundaryIndices(neighbour_boundary);
+            gamm_v = neighbour_scheme.getBoundaryBorrowing(neighbour_boundary);
+
+            u = obj;
+            v = neighbour_scheme;
+
+            b1_u = gamm_u*u.lambda(I_u)./u.a11(I_u).^2;
+            b2_u = gamm_u*u.lambda(I_u)./u.a22(I_u).^2;
+            b1_v = gamm_v*v.lambda(I_v)./v.a11(I_v).^2;
+            b2_v = gamm_v*v.lambda(I_v)./v.a22(I_v).^2;
+
+            tau1 = -1./(4*b1_u) -1./(4*b1_v) -1./(4*b2_u) -1./(4*b2_v);
+            tau1 = tuning * spdiag(tau1);
+            tau2 = 1/2;
+
+            sig1 = -1/2;
+            sig2 = 0;
+
+            tau = (e_u*tau1 + tau2*d_u)*H_b_u;
+            sig = (sig1*e_u + sig2*d_u)*H_b_u;
+
+            closure = obj.a*obj.Hi*( tau*e_u' + sig*d_u');
+            penalty = obj.a*obj.Hi*(-tau*e_v' + sig*d_v');
+        end
+
+        function [closure, penalty] = interfaceNonConforming(obj,boundary,neighbour_scheme,neighbour_boundary,type)
+
+            % TODO: Make this work for curvilinear grids
+            warning('LaplaceCurvilinear: Non-conforming grid interpolation only works for Cartesian grids.');
+
+            % User can request special interpolation operators by specifying type.interpOpSet
+            default_field(type, 'interpOpSet', @sbp.InterpOpsOP);
+            interpOpSet = type.interpOpSet;
+            tuning = type.tuning;
+
+
+            % u denotes the solution in the own domain
+            % v denotes the solution in the neighbour domain
+            [e_u, d_u] = obj.getBoundaryOperator({'e', 'd'}, boundary);
+            H_b_u = obj.getBoundaryQuadrature(boundary);
+            I_u = obj.getBoundaryIndices(boundary);
+            gamm_u = obj.getBoundaryBorrowing(boundary);
+
+            [e_v, d_v] = neighbour_scheme.getBoundaryOperator({'e', 'd'}, neighbour_boundary);
+            H_b_v = neighbour_scheme.getBoundaryQuadrature(neighbour_boundary);
+            I_v = neighbour_scheme.getBoundaryIndices(neighbour_boundary);
+            gamm_v = neighbour_scheme.getBoundaryBorrowing(neighbour_boundary);
+
+
+            % Find the number of grid points along the interface
+            m_u = size(e_u, 2);
+            m_v = size(e_v, 2);
+
+            Hi = obj.Hi;
+            a = obj.a;
+
+            u = obj;
+            v = neighbour_scheme;
+
+            b1_u = gamm_u*u.lambda(I_u)./u.a11(I_u).^2;
+            b2_u = gamm_u*u.lambda(I_u)./u.a22(I_u).^2;
+            b1_v = gamm_v*v.lambda(I_v)./v.a11(I_v).^2;
+            b2_v = gamm_v*v.lambda(I_v)./v.a22(I_v).^2;
+
+            tau_u = -1./(4*b1_u) -1./(4*b2_u);
+            tau_v = -1./(4*b1_v) -1./(4*b2_v);
+
+            tau_u = tuning * spdiag(tau_u);
+            tau_v = tuning * spdiag(tau_v);
+            beta_u = tau_v;
+
+            % Build interpolation operators
+            intOps = interpOpSet(m_u, m_v, obj.order, neighbour_scheme.order);
+            Iu2v = intOps.Iu2v;
+            Iv2u = intOps.Iv2u;
+
+            closure = a*Hi*e_u*tau_u*H_b_u*e_u' + ...
+                      a*Hi*e_u*H_b_u*Iv2u.bad*beta_u*Iu2v.good*e_u' + ...
+                      a*1/2*Hi*d_u*H_b_u*e_u' + ...
+                      -a*1/2*Hi*e_u*H_b_u*d_u';
+
+            penalty = -a*Hi*e_u*tau_u*H_b_u*Iv2u.good*e_v' + ...
+                      -a*Hi*e_u*H_b_u*Iv2u.bad*beta_u*e_v' + ...
+                      -a*1/2*Hi*d_u*H_b_u*Iv2u.good*e_v' + ...
+                      -a*1/2*Hi*e_u*H_b_u*Iv2u.bad*d_v';
+
+        end
+
+        % Returns the boundary operator op for the boundary specified by the string boundary.
+        % op        -- string or a cell array of strings
+        % boundary  -- string
+        function varargout = getBoundaryOperator(obj, op, boundary)
+
+            if ~iscell(op)
+                op = {op};
+            end
+
+            for i = 1:numel(op)
+                switch op{i}
+                case 'e'
+                    switch boundary
+                    case 'w'
+                        e = obj.e_w;
+                    case 'e'
+                        e = obj.e_e;
+                    case 's'
+                        e = obj.e_s;
+                    case 'n'
+                        e = obj.e_n;
+                    otherwise
+                        error('No such boundary: boundary = %s',boundary);
+                    end
+                    varargout{i} = e;
+
+                case 'd'
+                    switch boundary
+                    case 'w'
+                        d = obj.d_w;
+                    case 'e'
+                        d = obj.d_e;
+                    case 's'
+                        d = obj.d_s;
+                    case 'n'
+                        d = obj.d_n;
+                    otherwise
+                        error('No such boundary: boundary = %s',boundary);
+                    end
+                    varargout{i} = d;
+                end
+            end
+        end
+
+        % Returns square boundary quadrature matrix, of dimension
+        % corresponding to the number of boundary points
+        %
+        % boundary -- string
+        function H_b = getBoundaryQuadrature(obj, boundary)
+
+            switch boundary
+                case 'w'
+                    H_b = obj.H_w;
+                case 'e'
+                    H_b = obj.H_e;
+                case 's'
+                    H_b = obj.H_s;
+                case 'n'
+                    H_b = obj.H_n;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+        end
+
+        % Returns square boundary quadrature scaling matrix, of dimension
+        % corresponding to the number of boundary points
+        %
+        % boundary -- string
+        function s_b = getBoundaryScaling(obj, boundary)
+
+            switch boundary
+                case 'w'
+                    s_b = obj.s_w;
+                case 'e'
+                    s_b = obj.s_e;
+                case 's'
+                    s_b = obj.s_s;
+                case 'n'
+                    s_b = obj.s_n;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+        end
+
+        % Returns the coordinate number corresponding to the boundary
+        %
+        % boundary -- string
+        function m = getBoundaryNumber(obj, boundary)
+
+            switch boundary
+                case {'w', 'e'}
+                    m = 1;
+                case {'s', 'n'}
+                    m = 2;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+        end
+
+        % Returns the indices of the boundary points in the grid matrix
+        % boundary -- string
+        function I = getBoundaryIndices(obj, boundary)
+            ind = grid.funcToMatrix(obj.grid, 1:prod(obj.m));
+            switch boundary
+                case 'w'
+                    I = ind(1,:);
+                case 'e'
+                    I = ind(end,:);
+                case 's'
+                    I = ind(:,1)';
+                case 'n'
+                    I = ind(:,end)';
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+        end
+
+        % Returns borrowing constant gamma
+        % boundary -- string
+        function [theta_H, theta_M, theta_R] = getBoundaryBorrowing(obj, boundary)
+            switch boundary
+                case {'w','e'}
+                    theta_H = obj.theta_H_u;
+                    theta_M = obj.theta_M_u;
+                    theta_R = obj.theta_R_u;
+                case {'s','n'}
+                    theta_H = obj.theta_H_v;
+                    theta_M = obj.theta_M_v;
+                    theta_R = obj.theta_R_v;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+        end
+
+        function N = size(obj)
+            N = prod(obj.m);
+        end
+    end
+end