How much does a "converged" answer typically differ from the true continuum solution?

A convergence study only tells you the answer has stopped changing much with further refinement — it doesn't independently prove that value equals the true continuum answer, though in well-behaved (non-singular) problems the two are typically very close once change between successive meshes is a few percent or less. Comparison against a hand-calculation, a closed-form solution, or physical test data remains the gold-standard check when one is available.

Is a finer mesh always safe, just potentially wasteful?

Usually, but not always. Beyond the singular-stress case (where refinement never converges and instead diverges toward infinity), extremely fine meshes can also introduce numerical issues like ill-conditioned matrices or excessive round-off accumulation, and in nonlinear/contact problems can sometimes worsen convergence of the nonlinear solver itself even as spatial accuracy improves. In practice the far more common failure is the opposite — meshes that are too coarse — but "just make it finer" is not an unconditional safety net.

Do I need a convergence study for every single FEA run?

Not for every run, but for any run whose result is driving a real engineering decision — a pass/fail against a stress allowable, a fatigue life prediction, a go/no-go on a design change. For quick comparative studies between design variants using the same mesh strategy consistently, the absolute convergence of any one variant matters less than whether the comparison between variants is fair, though a convergence check on at least the baseline is still good practice.

What is the difference between h-refinement and p-refinement?

H-refinement (the type discussed above) makes elements smaller while keeping their polynomial order fixed. P-refinement instead keeps element size fixed but raises the polynomial order of the shape functions within each element, which can converge faster for smooth solutions but is less commonly exposed as a simple user-facing option in mainstream CAD-integrated FEA tools than straightforward mesh refinement is.

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Mesh Density vs. Mesh Convergence

A finer mesh is a knob you turn. Convergence is the evidence that you turned it far enough — and only one of the two actually tells you whether to trust the result.

A common shortcut engineers take under deadline pressure is picking a mesh size that "looks fine enough" and running with it — a dense-looking mesh with small elements, an elevated element count, a nice smooth-shaded stress plot. None of that is evidence the answer is correct. Mesh density describes how fine the mesh is; mesh convergence describes what happens to your result of interest as you make it finer. A single mesh, however dense, gives you exactly one number. Convergence requires at least two or three meshes at increasing density and a check on whether the answer is settling down toward a stable value — which is the only way to know whether the mesh you used was actually fine enough to trust.