More Like this

1. Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

   https://doi.org/10.1090/mcom/3842  

   Wang, Haijin; Li, Fengyan; Shu, Chi-Wang; Zhang, Qiang (November 2023, Mathematics of Computation)

   In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the $L^2$norm of the numerical solution does not increase in time, under the time step condition $\mathrm{\tau <\#comment/>}\le <\#comment/>\mathcal{F}(h/c,d/c^2)$, with the convection coefficient $c$, the diffusion coefficient $d$, and the mesh size $h$. The function $\mathcal{F}$depends on the specific IMEX temporal method, the polynomial degree $k$of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes $\mathrm{\tau <\#comment/>}\lesssim <\#comment/>h/c$in the convection-dominated regime and it becomes $\mathrm{\tau <\#comment/>}\lesssim <\#comment/>d/c^2$in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions. 

   more » « less
2. A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

   https://doi.org/10.1090/mcom/3843  

   Hou, Dianming; Ju, Lili; Qiao, Zhonghua (November 2023, Mathematics of Computation)

   In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete $H^1$error estimate and energy stability for the classic constant mobility case and the $L^{\mathrm{\infty <\#comment/>}}$error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy. 

   more » « less
3. On the Size of the Singular Set of Minimizing Harmonic Maps

   https://doi.org/10.1090/memo/1519  

   Mazowiecka, Katarzyna; Miśkiewicz, Michał; Schikorra, Armin (October 2024, Memoirs of the American Mathematical Society)

   We consider minimizing harmonic maps $u$from $\mathrm{\Omega <\#comment/>}\subset <\#comment/>{\mathbb{R}}^n$into a closed Riemannian manifold $\mathcal{N}$and prove: 1. an extension to $n\ge <\#comment/>4$of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$is finite, we have\[ ${\mathcal{H}}^{n-<\#comment/>3}(\mathrm{sing}<\#comment/>u)\le <\#comment/>C{\int <\#comment/>}_{\mathrm{\partial <\#comment/>}\mathrm{\Omega <\#comment/>}}|{\mathrm{\nabla <\#comment/>}}_{T}u{|}^{n-<\#comment/>1}\mathrm{d}{\mathcal{H}}^{n-<\#comment/>1};$\]2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is $\mathcal{N}={\mathbb{S}}^2$we obtain that the singular set of $u$is stable under small $W^{1,n-<\#comment/>1}$-perturbations of the boundary data. In dimension $n=3$both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$with $s\in <\#comment/>(\frac{1}{2},1]$and $p\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$satisfying $sp\ge <\#comment/>2$. We also discuss sharpness. 

   more » « less
4. A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

   https://doi.org/10.1090/cams/39  

   Brenner, Susanne; Sung, Li-yeng; Tan, Zhiyu; Zhang, Hongchao (October 2024, Communications of the American Mathematical Society)

   We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in ${\mathbb{R}}^2$. It is based on an isoparametric $C^0$finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions.A priorianda posteriorierror estimates together with corroborating numerical results are presented. 

   more » « less
5. Holomorphic curves in the 6-pseudosphere and cyclic surfaces

   https://doi.org/10.1090/tran/9172  

   Collier, Brian; Toulisse, Jérémy (June 2024, Transactions of the American Mathematical Society)

   The space ${\mathbf{H}}^{4,2}$of vectors of norm $-<\#comment/>1$in ${\mathbb{R}}^{4,3}$has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form ${\mathsf{G}}_2^{\prime }$. In this paper we consider a class of holomorphic curves in ${\mathbf{H}}^{4,2}$which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group is naturally parameterized by certain ${\mathsf{G}}_2^{\prime }$-Higgs bundles. This leads to a holomorphic description of the moduli space as a fibration over Teichmüller space with a holomorphic action of the mapping class group. Using a generalization of Labourie’s cyclic surfaces, we then show that equivariant alternating holomorphic curves are infinitesimally rigid. 

   more » « less