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.

翻译：我们提出了一种新的$hp$-自适应策略，用于处理自伴椭圆边值问题，该方法不依赖于经典的后验误差估计子。相反，我们的方法基于一种普遍适用的能量误差缩减预测策略，该策略可通过底层离散逼近空间中自由度的局部修改来表达。与所提出的预测策略相关的计算涉及低维线性问题，这些问题计算成本低且高度可并行化。这一新概念的数学构建模块首先在抽象希尔伯特空间层面上进行开发，随后在$hp$型有限元离散化的具体背景下加以应用。针对这一特定框架，我们讨论了一种通过适当约束系数技术显式构造$p$-富化和$hp$-细化方法，该技术适用于任意维度。最后，通过一维和二维数值算例验证了所得$hp$-自适应策略的适用性和有效性。