Hybrid High-Order methods are introduced and analyzed for the elliptic obstacle problem in two and three space dimensions. The methods are formulated in terms of face unknowns which are polynomials of degree \(k=0\) or \(k=1\) and in terms of cell unknowns which are polynomials of degree \(l=0\). The discrete obstacle constraints are enforced on the cell unknowns. Higher polynomial degrees are not considered owing to the modest regularity of the exact solution. A priori error estimates of optimal order, that is, up to the expected regularity of the exact solution, are shown. Specifically, for \(k=1\), the method employs a local quadratic reconstruction operator and achieves an energy-error estimate of order \(h^{\frac{3}{2}-\epsilon }\), \(\epsilon >0\). To our knowledge, this result fills a gap in the literature for the quadratic approximation of the three-dimensional obstacle problem. Numerical experiments in two and three space dimensions illustrate the theoretical results.

介绍了混合高阶方法，并针对二维和三维空间中的椭圆障碍问题进行了分析。这些方法是根据面部未知数（即多项式\（k = 0 \）或\（k = 1 \））以及根据单元格未知数（即多项式\\ l = 1 \\）制定的。离散障碍约束在单元未知数上强制执行。由于精确解的适度规律性，因此不考虑较高的多项式次数。显示了最佳顺序的先验误差估计，即达到精确解的预期规律性。具体来说，对于\（k = 1 \），该方法采用局部二次重构算子并获得阶次的能量误差估计\（h ^ {\ frac {3} {2}-\ epsilon} \），\（\ epsilon> 0 \）。据我们所知，该结果填补了三维障碍问题的二次逼近的文献空白。在两个和三个空间维度上的数值实验说明了理论结果。