Nonconforming Virtual Element Method for 2m-th Order Partial Differential Equations in R^n with m>n
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The H^m-nonconforming virtual elements of any order k on any shape of polytope in R^n with constraints m> n and k≥ m are constructed in a universal way. A generalized Green's identity for H^m inner product m>n is derived, which is essential to devise the H^m-nonconforming virtual elements. The dimension of the H^m-nonconforming virtual elements can reduced by the Serendipity approach. By means of the local H^m projection and a stabilization term using the boundary degrees of freedom, the H^m-nonconforming virtual element methods are proposed to approximate solutions of the m-harmonic equation. The norm equivalence of the stabilization on the kernel of the local H^m projection is proved by using the bubble function technique, the Poincaré inquality and the trace inequality, which implies the well-posedness of the virtual element methods. Finally, the optimal error estimates for the H^m-nonconforming virtual element methods are achieved from an estimate of the weak continuity and the error estimate of the canonical interpolation.
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