This question might be silly. Is laplcae operator a positive definite
system and easy to solve? Does it become ill-conditioned when
mesh size and mesh quality reduce?
"ill-conditioned" and "difficult to solve" are two different things. The
condition number of the discretized Laplace operator (i.e., the Laplace
matrix) grows like O(1/h^2), so it becomes quite large on fine meshes. But
linear systems with this matrix are relatively easy to solve if you use
multigrid methods.
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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