The finite element method is an important, widely used numerical technique for solving partial differential equations. This technique utilizes basis functions for approximating the geometry and the variation of the solution field over finite regions, or elements, of the domain. These basis functions are generally formed by combinations of polynomials. In the past, the polynomial order of the basis has been low-typically of linear and quadratic order. However, in recent years so-called p and hp methods have been developed, which may elevate the order of the basis to arbitrary levels with the aim of accelerating the convergence of the numerical solution. The increasing complexity of numerical basis functions poses a significant challenge to visualization systems. In the past, such systems have been loosely coupled to simulation packages, exchanging data via file transfer, and internally reimplementing the basis functions in order to perform interpolation and implement visualization algorithms. However, as the basis functions become more complex and, in some cases, proprietary in nature, it becomes increasingly difficult if not impossible to reimplement them within the visualization system. Further, most visualization systems typically process linear primitives, in part to take advantage of graphics hardware and, in part, due to the inherent simplicity of the resulting algorithms. Thus, visualization of higher-order finite elements requires tessellating the basis to produce data compatible with existing visualization systems. In this paper, we describe adaptive methods that automatically tessellate complex finite element basis functions using a flexible and extensible software framework. These methods employ a recursive, edge-based subdivision algorithm driven by a set of error metrics including geometric error, solution error, and error in image space. Further, we describe advanced pretessellation techniques that guarantees capture of the critical points of the polynomial basis. The framework has been designed using the adaptor design pattern, meaning that the visualization system need not reimplement basis functions, rather it communicates with the simulation package via simple programmatic queries. We demonstrate our method on several examples, and have implemented the framework in the open-source visualization system VTK.

Techniques in numerical simulation such as the finite element method depend on basis functions for approximating the geometry and variation of the solution over discrete regions of a domain. Existing visualization systems can visualize these basis functions if they are linear, or for a small set of simple non-linear bases. However, newer numerical approaches often use basis functions of elevated and mixed order or complex form; hence existing visualization systems cannot directly process them. In this paper we describe an approach that supports automatic, adaptive tessellation of general basis functions using a flexible and extensible software architecture in conjunction with an on demand, edge-based recursive subdivision algorithm. The framework supports the use of functions implemented in external simulation packages, eliminating the need to reimplement the bases within the visualization system. We demonstrate our method on several examples, and have implemented the framework in the open-source visualization system VTK.