On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

摘要

We consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial-boundary conditions. Our estimation technique uses a full-discrete numerical scheme, which is based on the Galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space. In our technique, the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role. Combining these estimates with an argument for the discretized inverse operator and a contraction property of theNewtontype formulation, we derive an a posteriori estimate of the norm for the infinitedimensional operator. In numerical examples, we show that the proposed method should bemore efficient than the existingmethod. Moreover, as an application, we give some prototype results for numerical verification of solutions of nonlinear parabolic problems, which confirm the actual usefulness of our technique.