In this work, we have designed conforming and nonconforming virtual element methods (VEM) to approximate non-stationary nonlocal biharmonic equation on general shaped domain. By employing Faedo–Galerkin technique, we have proved the existence and uniqueness of the continuous weak formulation. Upon applying Brouwer’s fixed point theorem, the well-posedness of the fully discrete scheme is derived. Further, following [J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math.386 (2021) 113229], we have introduced Enrichment operator and derived a priori error estimates for fully discrete schemes on polygonal domains, not necessarily convex. The proposed error estimates are justified with some benchmark examples.

在这项工作中，我们设计了一致和非一致虚拟元素方法（VEM）来近似一般形状域上的非平稳非局部双调和方程。通过采用Faedo-Galerkin技术，我们证明了连续弱公式的存在性和唯一性。应用布劳威尔不动点定理，导出了完全离散格式的适定性。此外，遵循[J. Huang 和 Y. Yu，泊松和双调和方程的非相容虚元方法的中间误差分析，J. Comput。应用。数学。386 (2021) 113229]，我们引入了Enrichment算子并推导了先验多边形域上完全离散方案的误差估计，不一定是凸的。所提出的误差估计通过一些基准示例得到了证明。