I have a mesh, faces $F$, edges $E$, and vertices $V$, and I have a list of predefined level set contours. What algorithm can I use to construct contours in the most efficient manner?A plot of the contour is shown above. Lines with the same color have the same $z$ value. 
Given values on a mesh, what algorithm can I use to construct efficiently level set contours?
computational geometry;data visualization
It can be done by a loop over each element and then for each element a loop over the edges. Then for each edge determine the coordinate for a given contour if it cross the edge, i.e. check if the contour takes a value in the interval spanned by the vertices values. If the contour crosses two edges, then draw the a line between the coordinates between detected crossing points at the edges.Doing this for all elements will create the contours. It is also an option to refine to a finer mesh via interpolation on each triangle and then use a delaunay triangulations of the refined mesh and then apply the routine as described above for achieving a sufficiently fine resolution.