An $hp$-adaptive strategy based on locally predicted error reductions

Abstract: We introduce a new $hp$-adaptive strategy for self-adjoint elliptic boundary value problems that does not rely on using classical a posteriori error estimators. Instead, our approach is based on a generally applicable prediction strategy for the reduction of the energy error that can be expressed in terms of local modifications of the degrees of freedom in the underlying discrete approximation space. The computations related to the proposed prediction strategy involve low-dimensional linear problems that are computationally inexpensive and highly parallelizable. The mathematical building blocks for this new concept are first developed on an abstract Hilbert space level, before they are employed within the specific context of $hp$-type finite element discretizations. For this particular framework, we discuss an explicit construction of $p$-enrichments and $hp$-refinements by means of an appropriate constraint coefficient technique that can be employed in any dimensions. The applicability and effectiveness of the resulting $hp$-adaptive strategy is illustrated with some $1$- and $2$-dimensional numerical examples.