a finite element mesh

To estimate the area of a lake on a map, you might cover it with a tiling of small triangles, compute each triangle's area, and add them up. The finite element mesh is that tiling, applied to a computational domain: the region where a PDE lives is chopped into a multitude of small, simple, non-overlapping pieces called elements, and the solution is approximated piece by piece on each one.

An element is a basic shape — in 2D usually a triangle or quadrilateral, in 3D a tetrahedron or hexahedron — and the elements meet edge-to-edge and vertex-to-vertex, covering the domain without gaps or overlaps. The corners (and sometimes edge or interior points) are the NODES that carry the unknowns. Two big strengths follow. First, geometry flexibility: unstructured triangular and tetrahedral meshes can wrap around curved boundaries, aircraft wings, engine blocks, or arteries, where a regular finite-difference grid of squares would struggle. Second, local refinement: you can pack many small elements where the solution changes fast (a stress concentration, a boundary layer) and use coarse elements where it is calm, spending effort only where it buys accuracy.

Mesh QUALITY matters as much as mesh fineness, and this is the honest pitfall. Long, thin 'sliver' triangles or badly distorted elements inflate the condition number of the stiffness matrix and ruin accuracy even if the elements are small — the error bounds depend on the smallest angle, not just the size h. Generating a good mesh on a complicated 3D geometry is itself a hard, sometimes semi-manual engineering task, and for many real simulations meshing consumes more human time than solving. Mesh generation, smoothing, and quality metrics (aspect ratio, minimum angle, skewness) are a serious sub-field of their own.