A new error analysis of nonconforming Bergan's energy-orthogonal element for the Extended Fisher-Kolmogorov equation

A nonconforming finite element method (FEM for short) is proposed and analyzed for the fourth order Extended Fisher-Kolmogorov (EFK for short) equation by employing the Bergan's energy-orthogonal plate element. Because the shape function and its first derivatives of this element are discontinuous at the element's vertices, which is quite different from the conventional finite elements used in the existing literature, a series of novel approaches including some a priori bounds and interpolation function splitting are developed to present a new error analysis for deriving optimal estimates of order O(h) in energy norm for both the semi-discrete and backward Euler fully-discrete schemes. At last, numerical experiments are also provided to verify the theoretical analysis.