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The variable-step deferred correction method constructs high-order schemes by modifying the low-order schemes, which not only relaxes the strict step-ratio constraints related to higher-order schemes, but also more accurately captures multi-scale ...$\mathrm{}{}_{}\mathrm{}$${}^{\mathrm{}}$

An adaptive time-stepping scheme is developed for the Zakharov-Rubenchik system to resolve the multiple time scales accurately and to improve the computational efficiency during long-time simulations. The Crank-Nicolson formula and the Fourier ...

We build an asymptotically compatible energy of the variable-step L2-$1_{\sigma }$ scheme for the time-fractional Allen–Cahn model with the Caputo’s fractional derivative of order $\alpha \in (0,1)$, under a weak step-ratio constraint ${\tau }_k/{\tau }_{k-1}\ge r_{\star }(\alpha )$ for $k\ge 2$, where ${\tau }_k$ is ...${}_{}\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}$${\mathrm{}}_{}$

In this paper, an effective fully-discrete implicit scheme for solving linear reaction-diffusion equations is constructed by using the variable-time-step two-step backward differentiation formula (VSBDF2) in time combining with the nonconforming ...${}^{\mathrm{}}{}^{\mathrm{}}$${}^{\mathrm{}}$${}^{\mathrm{}}$${}^{\mathrm{}}$$\mathrm{}{}_{}{}_{}\mathrm{}$${}_{}$${}_{}$${}_{}$${}^{\mathrm{}}$${}^{\mathrm{}}{}^{\mathrm{}}$${\mathrm{}}_{\mathrm{}}$${\mathrm{}}_{\mathrm{}}{}_{}$${}^{\mathrm{}}$${}^{\mathrm{}}{}^{\mathrm{}}$

Electrical impedance tomography (EIT) is a non-ionizing imaging modality suitable for bedside pulmonary imaging. Currently, lung imaging with EIT in the ICU is tomographic due to the need for real-time images. However, linearized reconstruction ...

In this paper, the convex-splitting BDF2 method with variable time-steps (proposed in Chen et al. SIAM J Numer Anal 57:495–525, 2019) is reconsidered for the Cahn–Hilliard model. We adopt the Fourier pseudo-spectral discretization in space. With ...$\mathrm{}{}_{}{}_{}{}_{\mathrm{}}\mathrm{}$${}^{\mathrm{}}$$\mathrm{}{}_{}\mathrm{}$

We study the, fully coupled, equations of quasistatic electroporoelasticity, and show that under a mild condition on the coupling parameter, a condition which is satisfied in practice, the equations of quasistatic electroporoelasticity have a ...

• A variational formulation for quasistatic electroporoelasticity is described.
• This is the first mathematically rigorous treatment of these equations.
• Under a condition on the coupling parameter the equations have a unique solution.

We analyze here a finite element space semidiscretization of the Allen-Cahn/Cahn-Hilliard system coupled with heat equation and based on the Maxwell-Cattaneo law. We prove that the semidiscrete solution converges weakly to the continuous solution ...

A variable-step L1 scheme is proposed for the time-fractional molecular beam epitaxy model without slope selection. By taking advantage of the convex splitting of nonlinear bulk, a sharp L 2 norm error estimate is established under a ...

A fully implicit two-step backward differentiation formula (BDF2) scheme with variable time steps is considered for solving the phase field crystal (PFC) model by combining with pseudo-spectral method in space. We show that the BDF2 scheme ...$\mathrm{}{}_{}{}_{}{}_{\mathrm{}}{}_{}\mathrm{}$

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn–Hilliard model. We focus on the numerical influences of time-step variations by using the ...${}^{\mathrm{}}$$\mathrm{}{}_{}{}_{}{}_{\mathrm{}}{}_{\mathrm{}}$${}_{\mathrm{}}$${}_{\mathrm{}}\mathrm{}$$\mathrm{}{}_{}{}_{\mathrm{}}$

Eighty years ago J.C. Jaeger et al. introduced a class of improper integrals, currently called “Jaeger integrals” that occur in theoretical models of diverse physical phenomena characterized by cylindrical geometry. One application ...

This paper proposes a new type of multilevel method for solving the Steklov eigenvalue problem based on Newton’s method. In this iteration method, solving the Steklov eigenvalue problem is replaced by solving a small-scale eigenvalue problem on ...

Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled ...

The backward differential formulas of order $\mathrm{k}$ (BDF-$\mathrm{k}$) for $3\le \mathrm{k}\le 5$ are analyzed for the molecular beam epitaxial (MBE) model without slope selection. We show that the fully implicit uniform BDF-$\mathrm{k}$ schemes are convex and uniquely solvable under a weak ...$\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}$${}^{\mathrm{}}$$\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}$${}^{\mathrm{}}$$\mathrm{}$

We address three related problems in the theory of elasticity, formulated in the framework of double forms: the Saint-Venant compatibility condition, the existence and uniqueness of solutions for equations arising in incompatible elasticity, and the ...

We prove that exponential moments of a fluctuation of the pure transport equation decay pointwisely almost as fast as $t^{-3}$ when the domain is any general strictly convex subset of $\mathbb{R}^3$ with the smooth boundary of the diffuse boundary ...

In this article, we propose an adaptive spectral graph wavelet method to solve partial differential equations on network-like structures using so-called spectral graph wavelets. The concept of spectral graph wavelets is based on the discrete graph ...

In this work, we propose a Crank--Nicolson-type scheme with variable steps for the time fractional Allen--Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense) and is maximum bound preserving. ...

We introduce new parametrized classes of shape admissible domains in $\mathbb{R}^n$, $n\geq 2$, and prove that they are compact with respect to the convergence in the sense of characteristic functions, the Hausdorff sense, the sense of compacts, and the ...