Generally we aim is to get a solution which is independent of the mesh.
However sometimes we have to find a compromise between a fine mesh with very high accuracy and coarser mesh with acceptable accuracy and acceptable computational cost.
Best regards.
To answer this question indirectly, have a look at Upper Riemann sum and Lower Riemann Sum in integration method. You can understand the gap easily, when you are moving from coarse mesh to finer mesh. Almost a similar technique is used in any numerical method which deals with PDE or ODE.
To my opinion, it depends upon number of Gauss Points you chose (or set to a default value in the software) for Integration. Depending upon the number of Gauss Points, your stiffness matrix can be less stiff (under-integration) or it can be too stiff (over-integration).
In short, it depends on Under-integration and Over-integration, whether your displacements will converge from below (i.e.,displacement value increases as you densify the mesh) or from above (i.e., displacement value decreases as you densify the mesh).
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