Dujin Zuo

Gang Chen, Daozhi Han, Jiaxuan Liu et al.

We propose a hybridizable discontinuous Galerkin (HDG) method combined with convex-concave splitting for the temporal discretization of the convective Cahn-Hilliard equation. The convection term is discretized explicitly without stabilization, yielding three key advantages: (1) unconditional stability, (2) preservation of the optimal convergence rate for piecewise constant approximations, and (3) a symmetric system after local elimination, enabling efficient solver via minimal residual methods. We establish optimal convergence rates in the $L^2$ norm for both the scalar and flux variables for any polynomial degree $k \geq 0$. To achieve optimal $L^2$-norm estimates, we introduce a specialized HDG elliptic projection operator and analyze its approximation properties. Within the HDG framework, local elimination is employed to reduce the degrees of freedom associated with the globally coupled unknowns, and the scalar variables exhibit superconvergence. Finally, numerical experiments validate the theoretical convergence rates and demonstrate the effectiveness of the proposed method.

Gang Chen, Yangwen Zhang, Dujin Zuo

We study mixed finite element/Crank--Nicolson discretizations of a nonlinear Oldroyd problem with general nonsingular and weakly singular memory kernels. Direct evaluation of the history term requires storing all previous velocity snapshots, which leads to $\mathcal{O}(mN)$ memory and $\mathcal{O}(mN^2)$ work over $N$ time steps, where $m$ denotes the number of spatial degrees of freedom. To reduce this burden, we compress the velocity history online by an incremental singular value decomposition and use the compressed representation in the discrete memory term. Under an approximate low-rank assumption of numerical rank $r$, the storage decreases to $\mathcal{O}((m+N)r)$, while the total history-evaluation work becomes $\mathcal{O}(mNr+rN^2)$. For nonsingular kernels, we derive a tolerance-dependent perturbation estimate showing that the baseline finite element accuracy is retained when the compression tolerance is sufficiently small. We also extend the approach to tempered weakly singular kernels via convolution quadrature. Numerical tests show near-indistinguishable solutions from the uncompressed scheme for the reported tolerances, together with substantial memory savings and reduced wall-clock time.