腾讯会议 ID: 287 899 062

For 2nd-order elliptic problem, we propose a C1 Petro-Galerkin method, in which kth-order $C^1$-conforming finite elements are adopted for the trial space, and k-2th order discontinuous ($C^{-1}$ or $L^2$) piecewise polynomials are used as the test space. This is in contrast to the classical $C^1$-conforming finite element methods when both trial and test spaces use $C^1$ continuous piecewise polynomials. There is another alternative, using $C^0$ continuous piecewise polynomials as the test space. However, both theoretical analysis and numerical test indicate that $C^1$-$L^2$ pair is superior to the $C^1$-$C^0$ pair in the Petrov-Galerkin method.

The advantage of the $C^1$-$L^2$ Petrov-Galekin method is that it approximates derivatives (gradient) much more accurately than its counterpart existing methods. We prove that at the element nodal points, numerical approximation for both function and its gradient converge at rate 2k-2. We also identify superconvergence points/lines inside elements for function, the first-order and second-order derivatives. Numerical test results demonstrate that our theoretical error bounds are sharp.