HIGH-ORDER NUMERICAL METHOD AND ERROR ANALYSIS BASED ON A MIXED SCHEME FOR FOURTH-ORDER PROBLEM IN A BALL

It is widely recognized that numerical computation in high-dimensional problems poses a significant challenge, particularly for intricate surface geometries, such as spherical and cylindrical domains. Therefore, in this paper, we present a high-precision numerical method based on a mixed scheme for solving fourth-order problems in a ball. The core concept behind this algorithm is to convert the original problem into a second-order coupled system by introducing an auxiliary Laplace equation. Subsequently, the second-order coupled system is disassembled into a sequence of one-dimensional decoupled secondorder problems through spherical harmonic function expansion and variable separation. Building on this foundation, we formulated their variational formulations and discretizations, and demonstrated the uniqueness of weak and approximate solutions, as well as provided an error estimate between them. Furthermore, we have extended the algorithm to accommodate general variable coefficients. Lastly, we present numerous numerical examples, confirming the theory's correctness and the algorithm's high precision.