Unique solvability and convergence analysis of the Lagrange multiplier approach for gradient flows

The unique solvability analysis and error estimate of the Lagrange multiplier approach for gradient flows is theoretically analyzed. We identify a necessary and sufficient condition that has to be satisfied for the nonlinear algebraic equation arising from the original Lagrange multiplier approach to admit a unique solution in the neighborhood of its exact solution. In turn, a modified Lagrange multiplier approach is proposed so that the computation can continue even if the aforementioned condition is not satisfied. Using Cahn-Hilliard equation as an example, we rigorously establish the unique solvability analysis and optimal error estimates of a second-order Lagrange multiplier scheme assuming this condition and that the time step size is sufficient small.

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