An LpL^p- Primal-Dual Weak Galerkin method for div-curl Systems

This paper presents a new $L^p$-primal-dual weak Galerkin (PDWG) finite element method for the div-curl system with the normal boundary condition for $p>1$. Two crucial features for the proposed $L^p$-PDWG finite element scheme are as follows: (1) it offers an accurate and reliable numerical solution to the div-curl system under the low $W^{\alpha, p}$-regularity ($\alpha>0$) assumption for the exact solution; (2) it offers an effective approximation of the normal harmonic vector fields on domains with complex topology. An optimal order error estimate is established in the $L^q$-norm for the primal variable where $\frac{1}{p}+\frac{1}{q}=1$. A series of numerical experiments are presented to demonstrate the performance of the proposed $L^p$-PDWG algorithm.Comment: 22 pages, 2 figures, 8 tables. arXiv admin note: text overlap with arXiv:2101.0346

Two types of spectral volume methods for 1-D linear hyperbolic equations with degenerate variable coefficients

In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. The two classes of SV methods are constructed by letting a piecewise $k$-th order ($k\ge 1$ is an arbitrary integer) polynomial function satisfy the local conservation law in each {\it control volume} obtained by dividing the interval element of the underlying mesh with $k$ Gauss-Legendre points (LSV) or Radaus points (RSV). The $L^2$-norm stability and optimal order convergence properties for both methods are rigorously proved for general non-uniform meshes. The superconvergence behaviors of the two SV schemes have been also investigated: it is proved that under the $L^2$ norm, the SV flux function approximates the exact flux with $(k+2)$-th order and the SV solution approximates the exact solution with $(k+\frac32)$-th order; some superconvergence behaviors at certain special points and for element averages have been also discovered and proved. Our theoretical findings are verified by several numerical experiments