부산대학교 도서관

The authors present an adaptive algorithm for one-dimensional elliptic boundary value problems solved with $hp$ finite elements. They prove that with the proposed scheme the error in the energy norm decreases monotonically at each step. Numerical experiments show that the decrease follows an exponential law, though a formal proof of optimality is still lacking. \par The method is based on a residual based a posteriori error estimator and on the use of a carefully selected set of refinement patterns. The adaptation procedure is formally extendable in higher dimension, though the cost of testing different refinement patterns could be high. No coarsening step has been considered in this work. Yet, the theoretical result leading to the monotone decrease of the error cannot be directly extended to higher dimensions since $p$-uniform equivalence between the discretisation error and the proposed residual-based estimation has been proved only in 1D.