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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]

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]

In recent years, a nonoverlapping Robin-type domain decomposition method (DDM) for the finite element discretization systems of the second order elliptic equations, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed since it was first proposed by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in Proceedings of the 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1990, pp. 202-223]. However, the convergence rate of this DDM with many subdomains remains open when the lower term of equations vanishes. This open problem will be considered in this paper. The convergence rate is almost 1 — O(h¹/²H-1/²) in certain cases—for example, the case of a small number of subdomains, where h is the mesh size and H is the size of subdomain. In order to get the desirous convergence results, two mathematics skills are introduced in this paper; one is complexification of real linear space and the other is the spectral radius formula. [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]

In this paper, we propose a nonoverlapping domain decomposition method for solving the three-dimensional Maxwell equations, based on the edge element discretization. For the Schur complement system on the interface, we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping sub domains. It is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

A nonoverlapping domain decomposition approach is used on uniform and matching grids to first define and then to compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on an L-shaped region. We prove existence and uniqueness of the collocation solution and derive optimal order Hs-norm error bounds for s = 0, 1, 2. The collocation solution on two interfaces is computed using the preconditioned conjugate gradient method, and the collocation solution on three squares is computed by a matrix decomposition method that uses fast Fourier transforms. The total cost of the algorithm is O(N² log N), where the number of unknowns in the collocation solution is O(N²). [ABSTRACT FROM AUTHOR]