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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]

In this work, we consider compressible single-phase flow problems in a porous medium containing a fracture. In the fracture, a nonlinear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a nonlinear interface problem. We then introduce two new algorithms that are able to efficiently handle the nonlinearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve for the nonlinearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by precomputing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann codimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings X and in particular, t. The stability and the convergence of both schemes are obtained, where user-given parameters are optimized to minimize the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we demonstrate a local convergence of an adaptive scalar solver which is practical for strongly diagonal dominant Jacobian problems such as in some systems of nonlinear equations arising from the application of a nonoverlapping domain decomposition method. The method is tested to a nonlinear interface problem of a multichip heat conduction problem. The numerical results show that the method performs slightly better than a Newton–Krylov method. [Copyright &y& Elsevier]