A partition of unity finite element method for nonlinear transient diffusion problems in heterogeneous materials

This work studies for the first time the solution of a nonlinear problem using an enriched finite element approach. Such problems can be highly demanding computationally. Hence, they can significantly benefit from the efficiency of the enriched finite elements. A robust partition of unity finite element method for solving transient nonlinear diffusion problems is presented. The governing equations include nonlinear diffusion coefficients and/or nonlinear source terms in both homogeneous and heterogeneous materials. To integrate the equations in time, we consider a linearly semi-implicit scheme in the finite element framework. As enrichment procedures, we consider a combination of exponential expansions to be injected in the finite element basis functions on coarse meshes. The proposed method shows a large reduction in the number of degrees of freedom required to achieve a fixed accuracy compared to the conventional finite element method. In addition, the proposed partition of unity method shows a stable behavior in treating both internal and external boundary layers in nonlinear diffusion applications. The performance of the proposed method is also used for the numerical simulation of heat conduction in functionally graded materials.