A posteriori error estimates and local adaptive steering for nonoverlapping domain decomposition in saddle-point mixed formulations
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We consider saddle-point mixed finite element discretizations of the model Darcy flow problem. We devise a nonoverlapping domain decomposition algorithm with the following specific features: 1) at each step of the algorithm, we give guaranteed and fully computable upper and lower bounds on the algebraic energy error; 2) we employ the estimates to steer the algorithm; 3) we combine a coarse grid solver, a subdomain Neumann solver, and a subdomain Dirichlet solver; 4) for each of the solvers, we use a line search to determine an optimal step size; 5) consequently, on each step, we obtain a Pythagorean formula for the error decrease, where the new algebraic energy error squared is the old algebraic energy error squared minus a computable error decrease (the lower a posteriori error estimate) squared; 6) the predicted spatial distribution of the algebraic energy error (per solver) enables for local solver adaptivity. Numerical experiments illustrate the theoretical developments.
