Multiscale finite element method for pde constrained optimization in high gradient problems

Abstract

A multiscale finite element method (MsFEM) was introduced for high gradient Partial Differential Equation (PDE) constrained optimization problem. Starting with the traditional fournode finite element method, additional nodes were inserted automatically at high gradient regions by an adaptive algorithm based on refinement criteria. A posteriori error estimation and error indicator were formulated. The error estimation was residual-based, while the error indicator was gradient-based. Using the information from the gradient-based error indicator, a p-refinement indicator was used to decide whether a given element should be refined or not via adaptive algorithm. Two sets of elements were used to design the adaptive algorithm: the regular elements and transition elements. The regular elements are the linear and quadratic elements, while the transition elements are the elements having both quadratic and linear sides, useful in transitioning from linear to quadratic elements during the implementation of the adaptive algorithm. The coupling resulted in a MsFEM. An exact solution containing high-gradient and multivariate polynomial functions that satisfies the PDE constraint and minimizes the objective function was also created using MAPLE software. A PDE constrained error analysis was also developed and implemented. The proposed MsFEM was applied to PDE constrained optimization problem with localised high gradient to analyse and validate the performance and accuracy of the proposed technique. The obtained numerical results from the analysis in terms of relative error showed an encouraging and promising performance of the scheme. The numerical results showed that the technique could help in solving high gradient problems with accuracy and minimum error.