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Adaptive dimension splitting algorithm for three-dimensional elliptic equations

An adaptive dimension splitting algorithm for three-dimensional (3D) elliptic equations is presented in this paper. We propose residual and recovery-based error estimators with respect to X−Y plane direction and Z direction, respectively, and construct the corresponding adaptive algorithm. Two-sided bounds of the estimators guarantee the efficiency and reliability of such error estimators. Numerical examples verify their efficiency both in estimating the error and in refining the mesh adaptively. This algorithm can be compared with or even better than the 3D adaptive finite element method with tetrahedral elements in some cases. What is more, our new algorithm involves only two-dimensional mesh and one-dimensional mesh in the process of refining mesh adaptively, and it can be implemented in parallel.