求解线性四阶抛物方程的全离散LBRFD-IPDG方法
A fully discrete LBRFD-IPDG method for linear fourth-order parabolic equations
中文总结 AI 辅助
针对带Dirichlet边界条件的线性四阶抛物方程，提出时间隐式LBRFD多步格式结合空间混合IPDG的全离散方法，通过理论分析得到误差阶，数值实验证实稳定性与最优收敛性。
中文摘要 AI 辅助
我们提出了一种求解带Dirichlet边界条件的线性四阶抛物方程的全离散方法，该方法在时间上采用隐式LBRFD多步格式，在空间上结合混合内罚间断Galerkin（IPDG，内部惩罚间断伽辽金）方法。时间离散化使用等距线性重心有理插值，并包含启动过程。为便于空间离散化，我们通过引入辅助变量对原问题进行重新表述。对于特定参数对$(n,d)$，LBRFD方法被证明是$A(α)$-稳定的，且比同阶的对应BDF$p$方法具有更宽的稳定角。我们通过$G$-能量技巧和离散Grönwall引理建立了稳定性和先验误差估计。理论分析得到总$L^2$误差估计为$h^{k-1}+τ^p$阶，其中当$n-d$为偶数时$p=d$，当$n-d$为奇数时$p=d+1$。空间收敛阶的降低归因于边界$\boldsymbol{∂Ω}$上的贡献。尽管有这一理论预测，数值实验仍证实了方法的稳定性，并展示出$h^{k+1}+τ^p$阶的最优收敛性。
英文摘要
We propose a fully discrete method for linear fourth-order parabolic equations with Dirichlet boundary conditions, combining an implicit LBRFD multistep scheme in time with a mixed interior penalty discontinuous Galerkin (IPDG) method in space. The temporal discretization employs equispaced linear barycentric rational interpolants and incorporates a startup procedure. To facilitate the spatial discretization, the original problem is reformulated through an auxiliary variable. For certain parameter pairs $(n,d)$, the LBRFD method is shown to be $A(α)$-stable and to possess a wider stability angle than the corresponding BDF$p$ method of the same order. Stability and a priori error estimates are established via a $G$-energy technique and the discrete Grönwall lemma. The theoretical analysis yields a total $L^2$ error estimate of order $h^{k-1}+τ^p$, where $p=d$ if $n-d$ is even and $p=d+1$ if $n-d$ is odd. The reduced spatial convergence rate is attributed to boundary contributions on $\partialΩ$. Despite this theoretical prediction, numerical experiments confirm the stability and demonstrate optimal convergence of order $h^{k+1}+τ^p$.
