Application of Adjoint Equations for Numerical Solution of Problems Using the Fictitious Domain Method

The article describes a method for increasing the accuracy of numerical results of mathematical physics problems solved using the fictitious domain method (FDM). The one-dimensional Burgers problem with a curved boundary is considered as a model problem. To demonstrate the effectiveness of the proposed FDM variant, the original domain is extended to a simple auxiliary domain and the coefficients and the right-hand side of the equation are extended accordingly. At the boundary of the auxiliary domain boundary conditions are set to facilitate the numerical solution of the auxiliary FDM problem. In the classic version of FDM, there are differences between numerical and exact solutions of problems. In order to reduce this difference at the boundary of the original region, this article uses the variational method. The application of the variational method leads to the solution of the conjugate problem of the auxiliary FDM problem. To minimize the functional difference of the boundary condition on the original boundary an iterative process of the conjugate gradient method is constructed. Numerous calculations have been carried out over a wide range of input parameters, which show the efficiency and high accuracy of the proposed numerical method.