New quadratic/serendipity finite volume element solutions on arbitrary triangular/quadrilateral meshes (Q6574219)

The paper addresses the development of new finite volume element (FVE) solutions to solve anisotropic diffusion equations on arbitrary triangular and quadrilateral meshes. \N\NThe scientific problem tackled in this paper revolves around improving the mathematical theory and application of finite volume element methods, particularly higher-order FVEs, for solving partial differential equations (PDEs) under challenging conditions, such as on arbitrary mesh configurations. The author notes that while the linear FVE method is well-established and stable on arbitrary triangular meshes, the development of higher-order methods, such as quadratic FVE schemes, has been limited by restrictive mesh conditions and assumptions. Specifically, existing quadratic FVE schemes often require certain mesh quality conditions, such as minimum angle constraints, which limit their applicability.\N\NTo address these limitations, the paper introduces a new postprocessing technique that enhances quadratic and serendipity finite element solutions. This method involves the construction of new quadratic dual meshes and the addition of specially designed bubble functions, ensuring that the postprocessed solutions satisfy local conservation properties on the new dual meshes. The innovation lies in proving the existence and uniqueness of these new solutions under very general conditions, including arbitrary triangular and convex quadrilateral meshes and full anisotropic diffusion tensors. This represents a significant advancement over previous methods, which were often limited to simpler mesh configurations or isotropic cases.\N\NThe methods employed involve constructing new dual meshes and bubble functions, followed by a postprocessing procedure that operates independently within each element. For triangular elements, the method involves solving a \(6 \times 6\) local linear algebraic system, while for quadrilateral elements, an \(8 \times 8\) system is solved. The author rigorously proves the solvability of these systems, ensuring that the postprocessed solutions maintain optimal convergence rates under \( H^1 \) and \( L^2 \) norms on the primal meshes. Additionally, the paper generalises previous work by extending it to arbitrary convex quadrilateral meshes and anisotropic diffusion equations, thereby broadening the applicability of these numerical methods.\N\NThe main findings of this research include the successful extension of quadratic FVE solutions to more general mesh configurations and anisotropic diffusion problems. The author demonstrates that the new solutions not only satisfy local conservation laws but also converge optimally to the exact solution under standard norms. The theoretical advancements are supported by numerical experiments that validate the proposed methods and confirm the accuracy and robustness of the new solutions.\N\NIn conclusion, this research significantly contributes to the field of numerical methods for PDEs by overcoming longstanding limitations associated with higher-order FVEs on arbitrary meshes. The introduction of new dual meshes and bubble functions, along with rigorous proofs of solution existence and uniqueness, positions this work as a valuable resource for researchers and practitioners seeking reliable numerical solutions for complex diffusion problems on non-standard meshes. The findings have the potential to impact various applied fields, including computational fluid dynamics and other areas where local conservation properties are crucial.