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Qingguang Guan

Publications and source records attributed to Qingguang Guan.

9 recordsLinked to original sources
arXiv2026

A Coupled Conforming-Nonconforming Galerkin Method for Poisson's Equation on Curved Domains

A coupled conforming-nonconforming Galerkin method is proposed for Poisson's equation on two-dimensional curved domains. The method applies a weak Galerkin discretization only on a thin boundary layer of curvilinear elements near the curved boundary, while using a standard continuous Galerkin discretization in the polygonal interior. In this way, geometric flexibility is retained where it is needed most, and the number of nonconforming degrees of freedom is significantly reduced. A key ingredient is a mixed interpolation-projection operator on curvilinear weak Galerkin elements, combining $L^2$ edge projections with conforming nodal interpolation on the interface side to ensure compatibility with the continuous Galerkin trace. Based on this construction, we prove an optimal a priori error estimate of order $O(h^k)$ in the energy norm under the basic geometric assumptions of the method, and an optimal $L^2(Ω)$ estimate of order $O(h^{k+1})$ under the additional elliptic dual regularity assumption. Numerical experiments confirm the theoretical rates and demonstrate substantial savings in degrees of freedom compared with a fully weak Galerkin discretization.

math.NA↗
arXiv2025

Unconditionally Stable, Variable Step DLN Methods for the Allen-Cahn Active Fluid Model: A Divergence-free Preserving Approach

This paper addresses the divergence-free mixed finite element method (FEM) for nonlinear fourth-order Allen-Cahn phase field coupled active fluid equations. By introducing an auxiliary variable $w = Δu$, the original fourth-order problem is converted into a system of second-order equations, thereby easing the regularity constraints imposed on standard $H^2$-comforming finite element spaces. To further refine the formulation, an additional auxiliary variable $ξ$, analogous to the pressure, is introduced, resulting in a mixed finite element scheme that preserves the divergence-free condition in $which = Δu$ inherited from the model. A fully discrete scheme is then established by combining the spatial approximation by the divergence-free mixed finite element method with the variable-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator. The boundedness of the scheme is rigorously derived under suitable regularity assumptions. Additionally, an adaptive time-stepping strategy based on the minimum dissipation criterion is carried out to enhance computational efficiency. Several numerical experiments validate the theoretical findings and demonstrate the method's effectiveness and accuracy in simulating complex active fluid dynamics.

math.NA↗
arXiv2025

A Divergence-free Preserving Mixed Finite Element Method for Thermally Driven Active Fluid Model

In this report, we propose a divergence-free preserving mixed finite element method (FEM) for the system of nonlinear fourth-order thermally driven active fluid equations. By introducing two auxiliary variables, we lower the complexity of the model and enhance the robustness of the algorithm. The auxiliary variable $w = Δu$ is used to convert the original fourth-order system to an equivalent system of second-order equations, thereby easing the regularity constraints imposed on standard $H^2$-conforming finite element space. The second variable $η$, analogous to the pressure, helps the scheme preserve the divergence-free condition arising from the model. The two-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator, unconditionally non-linear stable and second-order accurate under non-uniform time grids, is combined with the mixed FEM for fully discrete approximation. Due to the fine properties of the DLN scheme, we prove the boundedness of model energy and the associated error estimates under suitable regularity assumptions and mild time restrictions. Additionally, an adaptive time-stepping strategy based on a minimum-dissipation criterion is to balance computational costs and time efficiency. Several numerical experiments validate the theoretical findings and demonstrate the method's effectiveness and accuracy in simulating complex active fluid dynamics.

math.NA↗
arXiv2024

How Can Deep Neural Networks Fail Even With Global Optima?

Fully connected deep neural networks are successfully applied to classification and function approximation problems. By minimizing the cost function, i.e., finding the proper weights and biases, models can be built for accurate predictions. The ideal optimization process can achieve global optima. However, do global optima always perform well? If not, how bad can it be? In this work, we aim to: 1) extend the expressive power of shallow neural networks to networks of any depth using a simple trick, 2) construct extremely overfitting deep neural networks that, despite having global optima, still fail to perform well on classification and function approximation problems. Different types of activation functions are considered, including ReLU, Parametric ReLU, and Sigmoid functions. Extensive theoretical analysis has been conducted, ranging from one-dimensional models to models of any dimensionality. Numerical results illustrate our theoretical findings.

cs.LG↗
arXiv2024

Hybrid PDE-Deep Neural Network Model for Calcium Dynamics in Neurons

Traditionally, calcium dynamics in neurons are modeled using partial differential equations (PDEs) and ordinary differential equations (ODEs). The PDE component focuses on reaction-diffusion processes, while the ODE component addresses transmission via ion channels on the cell's or organelle's membrane. However, analytically determining the underlying equations for ion channels is highly challenging due to the complexity and unknown factors inherent in biological processes. Therefore, we employ deep neural networks (DNNs) to model the open probability of ion channels, a task that can be intricate when approached with ODEs. This technique also reduces the number of unknowns required to model the open probability. When trained with valid data, the same neural network architecture can be used for different ion channels, such as sodium, potassium, and calcium. Furthermore, based on the given data, we can build more physiologically reasonable DNN models that can be customized. Subsequently, we integrated the DNN model into calcium dynamics in neurons with endoplasmic reticulum, resulting in a hybrid model that combines PDEs and DNNs. Numerical results are provided to demonstrate the flexibility and advantages of the PDE-DNN model.

math.NA↗
arXiv2023

Weak Galerkin finite element method for second order problems on curvilinear polytopal meshes with Lipschitz continuous edges or faces

In this paper, we propose new basis functions defined on curved sides or faces of curvilinear elements (polygons or polyhedrons with curved sides or faces) for the weak Galerkin finite element method. Those basis functions are constructed by collecting linearly independent traces of polynomials on the curved sides/faces. We then analyze the modified weak Galerkin method for the elliptic equation and the interface problem on curvilinear polytopal meshes with Lipschitz continuous edges or faces. The method is designed to deal with less smooth complex boundaries or interfaces. Optimal convergence rates for $H^1$ and $L^2$ errors are obtained, and arbitrary high orders can be achieved for sufficiently smooth solutions. The numerical algorithm is discussed and tests are provided to verify theoretical findings.

math.NA↗
arXiv2022

Modeling calcium dynamics in neurons with endoplasmic reticulum: existence, uniqueness and an implicit-explicit finite element scheme

Like many other biological processes, calcium dynamics in neurons containing an endoplasmic reticulum are governed by diffusion-reaction equations on interface-separated domains. Interface conditions are typically described by systems of ordinary differential equations that provide fluxes across the interfaces. Using the calcium model as an example of this class of ODE-flux boundary interface problems, we prove the existence, uniqueness and boundedness of the solution by applying comparison theorem, fundamental solution of the parabolic operator and a strategy used in Picard's existence theorem. Then we propose and analyze an efficient implicit-explicit finite element scheme which is implicit for the parabolic operator and explicit for the nonlinear terms. We show that the stability does not depend on the spatial mesh size. Also the optimal convergence rate in $H^1$ norm is obtained. Numerical experiments illustrate the theoretical results.

math.NA↗
arXiv2021

Some Estimates of Virtual Element Methods for Fourth Order Problems

In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis is based on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates are obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions are provided.

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