A High-order Coupled Compact Integrated RBF Approximation Based Domain Decomposition Algorithm for Second-order Differential Problems

This paper presents a high-order coupled compact integrated RBF (CC IRBF) approximation based domain decomposition (DD) algorithm for the discretisation of second-order differential problems. Several Schwarz DD algorithms, including one-level additive/ multiplicative and two-level additive/ multiplicative/ hybrid, are employed. The CCIRBF based DD algorithms are analysed with different mesh sizes, numbers of subdomains and overlap sizes for Poisson problems. Our convergence analysis shows that the CCIRBF two-level multiplicative version is the most effective algorithm among various schemes employed here. Especially, the present CCIRBF two-level method converges quite rapidly even when the domain is divided into many subdomains, which shows great promise for either serial or parallel computing. For practical tests, we then incorporate the CCIRBF into serial and parallel two-level multiplicative Schwarz. Several numerical examples, including those governed by Poisson and Navier-Stokes equations are analysed to demonstrate the accuracy and efficiency of the serial and parallel algorithms implemented with the CCIRBF. Numerical results show: (i) the CCIRBF-Serial and -Parallel algorithms have the capability to reach almost the same solution accuracy level of the CCIRBF-Single domain, which is ideal in terms of computational calculations; (ii) the CCIRBF-Serial and -Parallel algorithms are highly accurate in comparison with standard finite difference, compact finite difference and some other schemes; (iii) the proposed CCIRBF-Serial and -Parallel algorithms may be used as alternatives to solve large-size problems which the CCIRBF-Single domain may not be able to deal with. The ability of producing stable and highly accurate results of the proposed serial and parallel schemes is believed to be the contribution of the coarse mesh of the two-level domain decomposition and the CCIRBF approximation. It is noted that the focus of this paper is on the derivation of highly accurate serial and parallel algorithms for second-order differential problems. The scope of this work does not cover a thorough analysis of computational time.