A new approach for the increase in the order of accuracy of high order elements used for the time dependent heat equation and for the time independent Poisson equation has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error. By a simple modification of the coefficients of 25-point stencils, the new approach exceeds the accuracy of the quadratic isogeometric elements by four orders for the heat equation and by twelve orders for the Poisson equation. Despite the significant increase in accuracy, the computational costs of the new technique are the same as those for the conventional quadratic isogeometric elements on a given mesh. The numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new approach is much more accurate than the conventional isogeometric elements at the same number of degrees of freedom. Hybrid methods that combine the new stencils with the conventional isogeometric and finite elements and can be applied to irregular domains are also presented.

在均匀的正方形和矩形网格上，已经提出了一种新的方法，该方法用于提高用于与时间相关的热方程和与时间无关的泊松方程的高阶元素的精度。它基于相对于局部截断误差的相应离散模板方程系数的优化。通过简单地修改25点模具的系数，新方法将二次等几何元素的精度提高到热方程的四阶和泊松方程的十二阶。尽管精度有了显着提高，但是新技术的计算成本与给定网格上常规二次等几何元素的计算成本相同。数值算例与新方法的理论结果非常吻合，并且还表明，在相同的自由度下，新方法比传统的等几何元素精确得多。还提出了将新模板与常规等几何和有限元结合起来并可以应用于不规则区域的混合方法。