The multi-level hp-method for three-dimensional problems: Dynamically changing high-order mesh refinement with arbitrary hanging nodes

One main challenge of the hp-version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately. The multi-level hp-formulation–recently introduced for two-dimensional applications–aims at alleviating these difficulties without compromising the approximation quality. This is achieved by changing from the conventional refine-by-replacement approach to a refine-by-superposition idea. The current work shows that the multi-level hp-approach can be extended naturally to three-dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions. In this way, a three-dimensional hp-refinement scheme is formulated, in which hanging nodes are avoided by definition. This ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime. Different numerical examples demonstrate that–even in the presence of singularities–this novel refinement scheme yields exponential convergence with respect to both, the number of unknowns and the computational time. It is further shown that the refinement scheme is able to capture complex solution features that demand for three-dimensional refinement patterns. The dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity. Finally, it is shown that the high approximation power of the multi-level hp-scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix.