Multiscale hybrid finite element and finite volume method for high gradient boundary value problems

Abstract

A multiscale hybrid finite element and finite volume method (MSHFEFVM) was introduced for high gradient boundary value problems by coupling an adaptive finite element and node centred finite volume schemes. Starting with the traditional four-node 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 which are 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. These elements are useful in transitioning from linear to quadratic elements during the implementation of the adaptive algorithm. The coupling resulted in a multiscale finite element method (MSFEM). The MSFEM was applied to some two-dimensional high gradient problems with promising results. The MSFEM was extended to solve the time dependent partial differential problems. The results obtained showed good agreement with the analytical results. A node centred finite volume method was coupled with the MSFEM to form a MSHFEFVM based on concurrent continuum-continuum coupling using a handshake coupling technique that allows information passing between the two coupled methods on a fly. The proposed hybrid technique was first applied to some two-dimensional localised high gradient problems with available analytical solutions. This application was necessary to analyse and validate the performance and accuracy of the MSHFEFVM. The obtained numerical results from the analysis in terms of error and execution time showed an encouraging performance of the scheme compared to the traditional finite element, the node centred finite volume and the MSFEM. Finally, the MSHFEFVM was applied to two standard localised high gradient problems and two engineering problems, which are electrostatics and torsion problems. The application showed a promising performance of the new scheme. The numerical results show that the combination of these two techniques can help to solve high gradient problems with accuracy and minimum execution time.