Splitting Extrapolation Method for Solving Second-order Parabolic Equations with Curved Boundaries by Using Domain Decomposition and D-quadratic Isoparametric Finite Elements

This article discusses a splitting extrapolation method for solving second-order parabolic equations with curved boundaries by using domain decomposition and d-quadratic isoparametric finite elements. This method possesses superconvergence, a high order of accuracy and a high degree of parallelism. First, we prove the multi-variable asymptotic expansion of fully discrete d-quadratic isoparametric finite element errors. Based on the expansion, we generate splitting extrapolation formulas. These formulas generate a numerical solution on a globally fine grid with higher accuracy by solving only a set of smaller discrete subproblems on different coarser grids. Therefore, a large-scale multidimensional problem with a curved boundary is turned into a set of smaller discrete subproblems on a polyhedron. Because these subproblems are independent of each other and have similar scales, our algorithm possesses a high degree of parallelism. In addition, this method is effective for solving discontinuous problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition. Our numerical results also show that the algorithm is effective for solving nonlinear parabolic equations.