In the finite element method, poor quality elements typically increase the condition number of the underlying stiffness matrix, thereby potentially: (1) degrading the computed solution, and (2) slowing the convergence of iterative solvers. Current mesh improvement strategies rely on node movement and edge-flipping to alleviate these problems. However, these methods cannot guarantee a lower-bound on mesh quality, especially in 3-D. In this paper we propose the concept, and use, of inverted elements to improve mesh quality and condition number. Inverted elements are standard finite elements, but with negative Jacobian. After establishing the mathematical properties of these elements we show how they can be used to dramatically improve the quality of a mesh through the use of an 'element cover'. Further, we show that a lower-bound on the mesh quality can be easily achieved, as supported by numerical experiments and case-studies.
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