You are an expert in writing code in many different languages. Your goal is to perform code completion for the following code, keeping in mind the rest of the code and the file meta data.
    Metadata details:
    filename: solveLaplace.m
    path of file: ./repos/ameshref/example3/P1AFEM_irr
    
    the code of the file until where you have to start completion: function [x,energy] = solveLaplace(coordinates,elements,constrains, ...
                                   dirichlet,neumann,f,g,uD)
%solveLaplace: computes P1-finite element solution for the two dimensional
%              Laplace equation with mixed Dirichlet-Neumann boundary 
%              condition
%
%    solveLaplace solves Laplace equation 
%      - div(grad(u)) = f                   in Omega
%                   u = u_D                 on the Dirichlet boundary
%              d/dn u = g                   on the Neumann boundary
%    on a geometry described by triangles with hanging nodes. 
%
%Usage:
%
%[x,energy] = solveLaplace(coordinates,elements,dirichlet,neumann,f,g,ud)
%
%Comments:
%
%    solveLaplace expects as input a finite element mesh described by the 
%    fields coordinates, elements,constrains, dirichlet and neumann. The volume
%    force f, the Neumann data g, and the (inhomogeneous) Dirichlet data
%    uD are given as M-files <f.m>, <g.m>, and <uD.m>. Either of these 
%    M-files is assumed to take n evaluation points as (n x 2) matrix and to
%    return an (n x 1) column vector. 
%
%    solveLaplace assembles the Galerkin data and solves the resulting 
%    linear system of equations to obtain the P1 finite element solution 
%    of the Laplace problem. The function returns a column vector X which
%    contains the nodal values of the FEM solution. Additionally, solveLaplace 
%    provides the energy of the discrete solution uh, i.e. 
%    energy = || grad(uh) ||_{L2(Omega)}^2.
%
%Remark:
%
%    This program is a supplement to the paper 
%    >> Adaptive Mesh Refinement in 2D - An Efficient Implementation in Matlab <<
%    by S. Funken, and A. Schmidt. The reader should 
%    consult that paper for more information.   
%
%Authors:
% 
%    S. Funken, A.Schmidt  21-08-18

nE = size(elements,1);
nC = size(coordinates,1);
x = zeros(nC,1);
%*** First vertex of elements and corresponding edge vectors 
c1 = coordinates(elements(:,1),:);
d21 = coordinates(elements(:,2),:) - c1;
d31 = coordinates(elements(:,3),:) - c1;
%*** Vector of element areas 4*|T|
area4 = 2*(d21(:,1).*d31(:,2)-d21(:,2).*d31(:,1));
%*** Assembly of stiffness matrix

    
function [x,energy] = solveLaplace(coordinates,elements,constrains, ...
                                   dirichlet,neumann,f,g,uD)
%solveLaplace: computes P1-finite element solution for the two dimensional
%              Laplace equation with mixed Dirichlet-Neumann boundary 
%              condition
%
%    solveLaplace solves Laplace equation 
%      - div(grad(u)) = f                   in Omega
%                   u = u_D                 on the Dirichlet boundary
%              d/dn u = g                   on the Neumann boundary
%    on a geometry described by triangles with hanging nodes. 
%
%Usage:
%
%[x,energy] = solveLaplace(coordinates,elements,dirichlet,neumann,f,g,ud)
%
%Comments:
%
%    solveLaplace expects as input a finite element mesh described by the 
%    fields coordinates, elements,constrains, dirichlet and neumann. The volume
%    force f, the Neumann data g, and the (inhomogeneous) Dirichlet data
%    uD are given as M-files <f.m>, <g.m>, and <uD.m>. Either of these 
%    M-files is assumed to take n evaluation points as (n x 2) matrix and to
%    return an (n x 1) column vector. 
%
%    solveLaplace assembles the Galerkin data and solves the resulting 
%    linear system of equations to obtain the P1 finite element solution 
%    of the Laplace problem. The function returns a column vector X which
%    contains the nodal values of the FEM solution. Additionally, solveLaplace 
%    provides the energy of the discrete solution uh, i.e. 
%    energy = || grad(uh) ||_{L2(Omega)}^2.
%
%Remark:
%
%    This program is a supplement to the paper 
%    >> Adaptive Mesh Refinement in 2D - An Efficient Implementation in Matlab <<
%    by S. Funken, and A. Schmidt. The reader should 
%    consult that paper for more information.   
%
%Authors:
% 
%    S. Funken, A.Schmidt  21-08-18

nE = size(elements,1);
nC = size(coordinates,1);
x = zeros(nC,1);
%*** First vertex of elements and corresponding edge vectors 
c1 = coordinates(elements(:,1),:);
d21 = coordinates(elements(:,2),:) - c1;
d31 = coordinates(elements(:,3),:) - c1;
%*** Vector of element areas 4*|T|
area4 = 2*(d21(:,1).*d31(:,2)-d21(:,2).*d31(:,1));
%*** Assembly of stiffness matrix
I = reshape(elements(:,[1 2 3 1 2 3 1 2 3])',9*nE,1);
J = reshape(elements(:,[1 1 1 2 2 2 3 3 3])',9*nE,1);
a = (sum(d21.*d31,2)./area4)';
b = (sum(d31.*d31,2)./area4)';
c = (sum(d21.*d21,2)./area4)';
A = [-2*a+b+c;a-b;a-c;a-b;b;-a;a-c;-a;c];
A = sparse(I,J,A(:));
%*** Prescribe values at Dirichlet nodes
fixed = unique(dirichlet);
x(fixed) = feval(uD,coordinates(fixed,:));
%*** Assembly of right-hand side
fsT = feval(f,c1+(d21+d31)/3);
b = accumarray(elements(:),repmat(12\area4.*fsT,3,1),[nC 1]);
if ~isempty(neumann)
    cn1 = coordinates(neumann(:,1),:);
    cn2 = coordinates(neumann(:,2),:);
    gmE = feval(g,(cn1+cn2)/2);
    b = b + accumarray(neumann(:),...
                      repmat(2\sqrt(sum((cn2-cn1).^2,2)).*gmE,2,1),[nC 1]);
end