Abstract We propose a high-order multistep projection method for the harmonic map heat flow from a bounded domain $$\varOmega \subset \mathbb {R}^d$$ Ω ⊂ R d into a given $$\mathcal {N}$$ N -dimensional smooth surface $$\Gamma \subset \mathbb {R}^{{{\mathcal {N}}}+1}$$ Γ ⊂ R N + 1 . At every time level, an auxiliary numerical solution is solved by a multistep backward difference formula with a mass-lumping finite element method in space, and then projected onto the surface $$\Gamma $$ Γ . The projected numerical solution is used in the backward difference formula and the extrapolation of nonlinearities in the following time levels. Such projection algorithms are convenient in computation while still preserving the pointwise geometric constraint of the solution to stay on the target surface $$\Gamma $$ Γ . The convergence of some low-order single-step projection algorithms based on the backward Euler and Crank–Nicolson schemes have been studied in many articles for harmonic map heat flow and related models into the unit sphere, while the convergence of high-order multistep projection methods still remains open. In this article, we propose a high-order multistep projection method for harmonic map heat flows into a general smooth surface (not necessarily the unit sphere) and prove its optimal-order convergence by combining four techniques, i.e., decomposition of the Nevanlinna–Odeh multiplier technique into approximately normal and tangential components separately, an almost orthogonal relation between the error functions associated to the auxiliary and projected numerical solutions, pointwise $$L^\infty $$ L ∞ error estimates, the use of orthogonal projection onto the target surface $$\Gamma $$ Γ . Numerical results are provided to support the theoretical analysis on the convergence of the high-order multistep projection methods.

Abstract In recent years, stochastic algorithms have been introduced to solve ill-posed inverse problems. These algorithms select a random subset of equations during each iteration, displaying excellent scalability and competitive performance in large-scale inverse problems. However, given the inherent ill-posed nature of the underlying problems and the presence of noise in measurement data, these algorithms often exhibit prominent oscillations and display a semi-convergence phenomenon, like all iterative regularization methods. This aspect poses challenges in obtaining an output with good approximation property. In this paper, by leveraging the spirit of the discrepancy principle we propose an a posteriori stopping rule for the stochastic mirror descent method for solving ill-posed inverse problems in Banach spaces. We show that the proposed stopping rule always terminates the method within a finite number of steps and the corresponding outcome converges toward the sought solution almost surely as the noise level approaches zero. Numerical simulations are reported to demonstrate the promising performance.

Abstract The perfectly matched layer method (PML method) is a truncation technique well known for the numerical treatment of wave scattering problems in unbounded domains. In this paper, we study the convergence of the PML method for the wave scattering from an open waveguide in $$\mathbb {R}^2_+=\{x\in \mathbb {R}^2:x_2>0\}$$ R + 2 = { x ∈ R 2 : x 2 > 0 } , where the refractive index is assumed to be a local perturbation of a function which is periodic with respect to $$x_1$$ x 1 and equal to one above a finite height. The problem is challenging from the theoretical, and also from the numerical, point of view due to the existence of guided waves. A typical way to deal with this difficulty is to apply the limiting absorption principle. Based on the Floquet-Bloch transform and a curve deformation theory, the solution, derived from the limiting absorption principle, is rewritten as the line integral (with respect to the Floquet-Bloch parameter) of the solution of a system of quasi-periodic problems. By comparing the Dirichlet-to-Neumann maps on a straight line above the locally perturbed periodic layer, we finally show that the PML method converges exponentially with respect to the PML parameter. Finally, some numerical examples are shown to illustrate the theoretical results.

Abstract In the present paper, we establish the well-posedness, stability, and (weak) convergence of a fully-discrete approximation of the unsteady $$p(\cdot ,\cdot )$$ p ( · , · ) -Navier–Stokes equations employing an implicit Euler step in time and a discretely inf-sup-stable finite element approximation in space. Moreover, numerical experiments are carried out that supplement the theoretical findings.

In this paper we analyze the numerical oscillations affecting loosely coupled schemes for hybrid-dimensional 0D–3D fluid–structure interaction (FSI) problems, which arise e.g. in the field of cardiovascular modeling, and we propose a novel stabilized scheme that cures this issue. We study several loosely coupled schemes, including the Dirichlet–Neumann (DN) and Neumann–Dirichlet (ND) schemes. In the first one, the 0D fluid model prescribes the pressure to the 3D structural mechanics model and receives the flow. In the second one, on the contrary, the fluid model receives the pressure and prescribes the flow. The terms DN and ND, employed in the FSI literature, are borrowed from domain decomposition methods, although here a single iteration is performed before moving on to the next time step (that is, the coupling is treated explicitly). Should the fluid be enclosed in a cavity, the DN scheme is affected by non-physical oscillations whose origin lies in the balloon dilemma, for which we provide an algebraic interpretation. Moreover, we show that also the ND scheme can be unstable for a range of parameter choices. Surprisingly, increasing either the viscous dissipation or the inertia of the structure favours the onset of oscillations and, for certain parameter choices, the ND is unconditionally unstable. In the presence of inertial terms, by reducing the time step size below a certain threshold, the amplitude of the numerical oscillations is even amplified. We provide an explanation for these facts and establish sharp stability bounds on the time step size. Our analysis extends to Robin–Robin schemes, based on linear combinations of the conditions of pressure continuity and either volume or flux continuity. While appropriate choices of Robin coefficients can achieve numerical stability, tuning these coefficients can be challenging in practice. To address these issues, we propose a numerically consistent stabilization term for the Neumann–Dirichlet scheme, inspired by physical insight on the onset of oscillations. We prove that our proposed stabilized scheme is absolutely stable for any choice of time step size. Notably, the proposed scheme does not require parameter tuning. These results are verified by several numerical tests. Finally, we apply the proposed stabilized scheme to an important problem in cardiac electromechanics, namely the coupling between a 3D cardiac model and a closed-loop lumped-parameter model of blood circulation. In this setting, our proposed scheme successfully removes the non-physical oscillations that would otherwise affect the numerical solution.