The Finite Element Method with Continuity Constraints for Stair-step Grids in Geoscience

This work investigates the enforcement of continuity constraints on stair-step grids which are specialized grids for simulations in geosciences. They are rectilinear in horizontal directions but locally discontinuous (i.e., nonconforming) in the vertical direction. Furthermore, they allow for (partly) collapsed elements in order to model pinched-out layers. A robust and efficient algorithm for enforcing continuity is proposed which is tailored to the special properties of stair-step grids. A number of two- and three-dimensional finite element method (FEM) simulations on stair-step grids are conducted. Thereby, the Lagrange multiplier and penalty method with different ansatz spaces are studied for pointwise and averaged constraints. A particulary useful choice is the penalty method with continuous constraints and penalty parameters that depend on the element size.