An unconditionally stable and L2 optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations

作者：Wu, Xiaoxin; Qiu, Weifeng; Pan, Kejia^*

来源：ADVANCES IN COMPUTATIONAL MATHEMATICS, 2023, 49(6): 83.

DOI：10.1007/s10444-023-10085-5

• Finite volume scheme
• Quadratic element
• Elliptic equation
• Inf-sup condition
• L-2 error estimate
• Triangular meshes

In this paper, we propose an unconditionally stable and L-2 optimal quadratic finite volume (FV) scheme for solving the two-dimensional anisotropic elliptic equation on triangular meshes. In quadratic FV schemes, the construction of the dual partition is closely related to the L-2 error estimate. While many dual partitions over triangular meshes have been investigated in the literature, only the one proposed by Wang and Li (SIAM J. Numer. Anal. 54:2729-2749, 2016) has been proven to achieve optimal L-2 norm convergence rate. This paper introduces a novel approach for constructing the dual partition using multiblock control volumes, which is also shown to optimally converge in the L-2 norm (O(h(3))). Furthermore, we present a new mapping from the trial space to the test space, which enables us to demonstrate that the inf-sup condition of the scheme holds independently of the minimal angle of the underlying mesh. To the best of our knowledge, this is the first unconditionally stable quadratic FV scheme over triangular meshes that achieves optimal L-2 norm convergence rate. We provide numerical experiments to validate our findings.