Multilayer coextrusion has become a popular commercial process for producing complex polymeric products from soda bottles to reflective coatings. Here we are interested in coextrusion of filled polymers to produce a capacitor by alternating conductive and dielectric layers. The filled materials can exhibit a viscosity and density mismatch, in addition to complex rheology. A numerical model of a multilayer coextrusion process is developed based on a finite element discretization and an Eulerian level set method to understand the moving boundary problem associated with the polymer-polymer interface. The goal of this work is to have a numerical capability suitable for optimizing and troubleshooting the coextrusion process, circumventing flow instabilities such as ribbing and barring, and reducing variability in layer thickness. Though these instabilities can be both viscous and elastic in nature, for this work a generalized Newtonian description of the fluid is used. Both Newtonian and shear-thinning constitutive equations are explored. Two different geometries are investigated with three-dimensional free surface models: 1) flow in the splitter, where the material is split and restacked to increase the number of layers and 2) Flow in the draw-down die where a square cross-section is downsized to a ribbon at the outflow. The results of this work show that modeling can be useful improve material choices and guide mold design.