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We prove convergence for the nonoverlapping Robin--Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov--Poincaré operators based on the p-structure and the Lp-generalization of the Lions--Magenes spaces. This framework allows the reformulation of the Robin--Robin method into a Peaceman--Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions. [ABSTRACT FROM AUTHOR]

We consider a second order elliptic problem with a heterogeneous coefficient written in mixed form. The nonoverlapping mortar domain decomposition method is efficient in parallel if the mortar interface coupling space has a restricted number of degrees of freedom. In the heterogeneous case, we define a new multiscale mortar space that incorporates purely local information from homogenization theory to better approximate the solution along the interfaces with just a few degrees of freedom. In the case of a locally periodic heterogeneous coefficient of period ε, we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when e is small. Moreover, we present three numerical examples to assess its performance when the coefficient is not obviously locally periodic. We show that the new mortar method works well, and better than polynomial mortar spaces. [ABSTRACT FROM AUTHOR]

A class of corrected explicit-implicit domain decomposition (CEIDD) methods is investigated for the parallel approximation of linear heat equations. Explicit-implicit domain decomposition (EIDD) methods are computationally and communicationally efficient for each time step but always suffer from small time step size restrictions. By adding an interface correction step to Kuznetsov's EIDD, the one-dimensional CEIDD procedure achieves unconditional stability without discarding the time-stepwise efficiency of the EIDD method. In order to maintain the virtues of the CEIDD method and improve the flexibility in domain partitioning, for solving multidimensional problems, special zigzag-shaped interfaces are suggested in the CEIDD method. Based on non-crossover and crossover types of zigzag interfaces, the resulting CEIDD-ZI algorithms are studied for two strategies of subdomain partition. By the energy method, it shows that the proposed algorithms, including their degenerate cases—the corrected explicit hopscotch schemes-are convergent in the discrete H¹ seminorm and L² norm. Numerical experiments confirm the results in our analysis. [ABSTRACT FROM AUTHOR]