A C^1-conforming Petrov-Galerkin method for convection-diffusion equations and superconvergence ananlysis over rectangular meshes
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In this paper, a new C^1-conforming Petrov-Galerkin method for convection-diffusion equations is designed and analyzed. The trail space of the proposed method is a C^1-conforming ℚ_k (i.e., tensor product of polynomials of degree at most k) finite element space while the test space is taken as the L^2 (discontinuous) piecewise ℚ_k-2 polynomial space. Existence and uniqueness of the numerical solution is proved and optimal error estimates in all L^2, H^1, H^2-norms are established. In addition, superconvergence properties of the new method are investigated and superconvergence points/lines are identified at mesh nodes (with order 2k-2 for both function value and derivatives), at roots of a special Jacobi polynomial, and at the Lobatto lines and Gauss lines with rigorous theoretical analysis. In order to reduce the global regularity requirement, interior a priori error estimates in the L^2, H^1, H^2-norms are derived. Numerical experiments are presented to confirm theoretical findings.
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