Towards fully mesh adaptive FE-simulations in 3D using multi-grid solver

The four cornerstones of efficient simulation in engineering practice are efficient and robust finite elements, solution of systems of linear equations, accuracy assessment (AA) and mesh generation and adaptation (MGA). This paper describes our current work in these fields as implemented in the in-house program FEM90. Special attention is given to indefinite systems due to constraint equations used to connect regions with non-conforming meshes, since this situation is commonly encountered in engineering structures. This type of system is especially problematic for iterative solvers, but the paper presents a new pre-conditioner that carry over the fast convergence of the pre-conditioned conjugate gradient method also to indefinite systems. Indeed, comparison tests of the direct and iterative solvers on realistic examples with up to 1,000,000 equations show that the iterative multi-grid solver has optimal performance and is superior both in terms of memory and time on large 3D problems. To solve even larger problems with required accuracy, the paper also implements and tests AA and MGA in 3D. The combination of the FEM90 and Ideas programs provides automatic refinement of both brick and tetrahedron meshes.