You may use the low-level patch function. Since I don't have your data, please find below a self-consistent example showing the difference between a plot of node-based function and an element-based function (obtained by aveaging the nod values).
The construction of the connectivity matrix may seem complex, but I should be simple to understand if you look at the matrix nodMap and then the corresponding T matrix.
clear variables, close all
% discretization params
nx = 5;
ny = 5;
% my data
x = linspace(0,1,nx);
y = linspace(0,1,ny);
[X,Y] = meshgrid(x,y);
P = [X(:) Y(:)];
% node map
nodMap = reshape(1:nx*ny,nx,ny);
% connectivity (4 nodes defining a rectangle)
T = zeros((nx-1)*(ny-1),4);
T(:,1) = reshape(nodMap(1:nx-1,1:ny-1),(nx-1)*(ny-1),1);
T(:,2) = reshape(nodMap(2:nx,1:ny-1),(nx-1)*(ny-1),1);
T(:,3) = reshape(nodMap(2:nx,2:ny),(nx-1)*(ny-1),1);
T(:,4) = reshape(nodMap(1:nx-1,2:ny),(nx-1)*(ny-1),1);
% my node-based function to plot f = x^2+y^2
fNod = P(:,1).^2+P(:,2).^2;
% my element-based function
fEle = sum(fNod(T),2)/size(T,2);
% plot node-based function
figure, patch('Vertices',P,'Faces',T,'FaceVertexCData',fNod,'EdgeColor','none','FaceColor','interp');
colorbar
% plot element-based function
figure, patch('Vertices',P,'Faces',T,'faceVertexCData',fEle,'EdgeColor','none','FaceColor','flat');
colorbar
