In this paper, we derive a priori error estimates for a class of interior penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE) functions for a classic second-order elliptic interface problem. The error estimation shows that these methods can converge optimally in a mesh-dependent energy norm. The combination of IFEs and DG formulation in these methods allows local mesh refinement in the Cartesian mesh structure for interface problems. Numerical results are provided to demonstrate the convergence and local mesh refinement features of these DG-IFE methods.

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We present partially penalized immersed finite element methods for solving parabolic interface problems on Cartesian meshes. Typical semidiscrete and fully discrete schemes are discussed. Error estimates in an energy norm are derived. Numerical examples are provided to support theoretical analysis.

In this article, interior penalty discontinuous Galerkin methods using immersed finite element functions are employed to solve parabolic interface problems. Typical semi-discrete and fully discrete schemes are presented and analyzed. Optimal convergence for both semi-discrete and fully discrete schemes are proved. Some numerical experiments are provided to validate our theoretical results.

Abstract:The thermo-elastic fracture problem and equations are established for aluminium alloy Metal Inert Gas (MIG) welding, which include a moving heat source and a thermoelasticity equation with the initial and boundary conditions for a plate structure with a crack. The extended finite element method (XFEM) is implemented to solve the thermo-elastic fracture problem of a plate structure with a crack under the effect of a moving heat source. The combination of the experimental measurement and simulation of the welding temperature field is done to verify the model and solution method. The numerical cases of the thermomechanical parameters and stress intensity factors (SIFs) of the plate structure in the welding heating and cooling processes are investigated. The research results provide reference data and an approach for the analysis of the thermomechanical characteristics of the welding process.