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A cubic \(H^3\)-nonconforming finite element. (English) Zbl 1449.65315

Summary: The lowest degree of polynomial for a finite element to solve a \(2k\) th-order elliptic equation is \(k\). The Morley element is such a finite element, of polynomial degree 2, for solving a fourth-order biharmonic equation. We design a cubic \(H^3\)-nonconforming macro-element on two-dimensional triangular grids, solving a sixth-order tri-harmonic equation. We also write down explicitly the 12 basis functions on each macro-element. A convergence theory is established and verified by numerical tests.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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