An hp-adaptive strategy based on locally predicted error reductions.
CoRR(2023)
摘要
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.
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