In this paper, we present a high-order finite element method based on a reconstructed approximation to the biharmonic equation. In our construction, the space is reconstructed from nodal values by solving a local least squares fitting problem per element. It is shown that the space can achieve an arbitrarily high-order accuracy and share the same nodal degrees of freedom with the $C^0$ linear space. The interior penalty discontinuous Galerkin scheme can be directly applied to the reconstructed space for solving the biharmonic equation. We prove that the numerical solution converges with optimal orders under error measurements. More importantly, we establish a norm equivalence between the reconstructed space and the continuous linear space. This property allows us to precondition the linear system arising from the high-order space by the linear space on the same mesh. This preconditioner is shown to be optimal in the sense that the condition number of the preconditioned system admits a uniform upper bound independent of the mesh size. Numerical examples in two and three dimensions are provided to illustrate the accuracy of the scheme and the efficiency of the preconditioning method.

翻译：本文提出了一种基于重构近似的高阶有限元方法用于求解双调和方程。在我们的构造中，空间通过在每个单元上求解局部最小二乘拟合问题，从节点值进行重构。研究表明，该空间能够达到任意高阶精度，并与$C^0$线性空间共享相同的节点自由度。内部罚间断伽辽金格式可直接应用于重构空间以求解双调和方程。我们证明了在误差度量下数值解以最优阶收敛。更重要的是，我们建立了重构空间与连续线性空间之间的范数等价性。这一性质使我们能够利用同一网格上的线性空间对高阶空间产生的线性系统进行预条件处理。该预条件器被证明是最优的，因为预条件系统的条件数具有与网格尺寸无关的一致上界。文中提供了二维和三维数值算例，以验证格式的精度和预条件方法的效率。