The Element Mesh Generation tutorial describes how good mesh accuracy is achieved through adjusting boundary nodes so they match the region boundary. This is illustrated with this example:
mesh2d = ToElementMesh[Disk]; \[Pi] - Total@First@mesh2d["MeshElementMeasure"] 2.00118*10^-6
When I try the 3d equivalent of this, I get a much bigger error:
mesh3d = ToElementMesh[Sphere]; 4/3 \[Pi] - Total@First@mesh3d["MeshElementMeasure"] 0.0213137
I suspect this node adjusting process works differently in 3d. Is there a way to improve the accuracy of 3d meshes without requiring a huge number of elements?