arXiv subjects

Dujin Zuo

Publications and source records attributed to Dujin Zuo.

3 recordsLinked to original sources

arXiv2026

A superconvergent hybridizable discontinuous Galerkin method for the convective Cahn--Hilliard equation

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.

math.NA↗

arXiv2026

Incremental SVD Compression for Nonlinear Oldroyd Equations with General Memory Kernels

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.

math.NA↗

arXiv2024

An Incremental SVD Method for Non-Fickian Flows in Porous Media: Addressing Storage and Computational Challenges

It is well known that the numerical solution of the Non-Fickian flows at the current stage depends on all previous time instances. Consequently, the storage requirement increases linearly, while the computational complexity grows quadratically with the number of time steps. This presents a significant challenge for numerical simulations. While numerous existing methods address this issue, our proposed approach stems from a data science perspective and maintains uniformity. Our method relies solely on the rank of the solution data, dissociating itself from dependency on any specific partial differential equation (PDE). In this paper, we make the assumption that the solution data exhibits approximate low rank. Here, we present a memory-free algorithm, based on the incremental SVD technique, that exhibits only linear growth in computational complexity as the number of time steps increases. We prove that the error between the solutions generated by the conventional algorithm and our innovative approach lies within the scope of machine error. Numerical experiments are showcased to affirm the accuracy and efficiency gains in terms of both memory usage and computational expenses.

math.NA↗