arXiv subjects

Yuxiang Liang

Publications and source records attributed to Yuxiang Liang.

4 recordsLinked to original sources

arXiv2025

Robust Augmented Mixed Finite Element Methods for Stoke Interface Problems with Discontinuous Viscosity in Multiple Subdomains

A stationary Stokes problem with a piecewise constant viscosity coefficient in multiple subdomains is considered in the paper. For standard finite element pairs, a robust inf-sup condition is required to show the robustness of the discretization error with respect to the discontinuous viscosity, which has only been proven for the two-subdomain case in the paper [Numer. Math. (2006) 103: 129--149]. To avoid the robust inf-sup condition of a discrete finite element pair for multiple subdomains, we propose an ultra-weak augmented mixed finite element formulation. By adopting a Galerkin-least-squares method, the augmented mixed formulation can achieve stability without relying on the inf-sup condition in both continuous and discrete settings. The key step to having a robust priori error estimate is to use two norms, one energy norm and one full norm, in robust continuity. The robust coercivity is proved for the energy norm. A robust a priori error estimate in the energy norm is then derived with the best approximation property in the full norm for the case of multiple subdomains. Additionally, the paper introduces a singular Kellogg-type example with exact solutions for the first time. Extensive numerical tests are conducted to validate the robust error estimate.

math.NA↗

arXiv2023

Least-Squares versus Partial Least-Squares Finite Element Methods: Robust A Priori and A Posteriori Error Estimates of Augmented Mixed Finite Element Methods

In this paper, for the generalized Darcy problem (an elliptic equation with discontinuous coefficients), we study a special partial Least-Squares (Galerkin-least-squares) method, known as the augmented mixed finite element method, and its relationship to the standard least-squares finite element method (LSFEM). Two versions of augmented mixed finite element methods are proposed in the paper with robust a priori and a posteriori error estimates. Augmented mixed finite element methods and the standard LSFEM uses the same a posteriori error estimator: the evaluations of numerical solutions at the corresponding least-squares functionals. As partial least-squares methods, the augmented mixed finite element methods are more flexible than the original LSFEMs. As comparisons, we discuss the mild non-robustness of a priori and a posteriori error estimates of the original LSFEMs. A special case that the $L^2$-based LSFEM is robust is also presented for the first time. Extensive numerical experiments are presented to verify our findings.

math.NA↗

arXiv2022

Least-Squares Methods with Nonconforming Finite Elements for General Second-Order Elliptic Equations

In this paper, we study least-squares finite element methods (LSFEM) for general second-order elliptic equations with nonconforming finite element approximations. The equation may be indefinite. For the two-field potential-flux div LSFEM with Crouzeix-Raviart (CR) element approximation, we present three proofs of the discrete solvability under the condition that mesh size is small enough. One of the proof is based on the coerciveness of the original bilinear form. The other two are based on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. A counterexample shows that div least-squares functional does not have norm equivalence in the sum space of $H^1$ and CR finite element spaces. Thus it cannot be used as an a posteriori error estimator. Several versions of reliable and efficient error estimators are proposed for the method. We also propose a three-filed potential-flux-intensity div-curl least-squares method with general nonconforming finite element approximations. The norm equivalence in the abstract nonconforming piecewise $H^1$-space is established for the three-filed formulation on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. The three-filed div-curl nonconforming formulation thus has no restriction on the mesh size, and the least-squares functional can be used as the built-in a posteriori error estimator. Under some restrictive conditions, we also discuss a potential-flux div-curl least-squares method.

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arXiv2019

A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation

In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L2-1{\sigma} formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H1-norm. Numerical examples are provided to justify the efficiency.

math.NA↗