Decay of weak solutions to Vlasov equation coupled with a shear thickening fluid

We show that the energy norm of weak solutions to Vlasov equation coupled with a shear thickening fluid on the whole space has a decay rate the energy norm $E(t) \leq {C}/{(1+t)^{\alpha }}, \forall t \geq 0$ for $\alpha \in (0,3/2)$ .

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

A Note on Stokes Approximations to Leray Solutions of the Incompressible Navier–Stokes Equations in ℝn

In the early 1980s it was well established that Leray solutions of the unforced Navier–Stokes equations in Rn decay in energy norm for large t. With the works of T. Miyakawa, M. Schonbek and others it is now known that the energy decay rate cannot in general be any faster than t−(n+2)/4 and is typically much slower. In contrast, we show in this note that, given an arbitrary Leray solution u(·,t), the difference of any two Stokes approximations to the Navier–Stokes flow u(·,t) will always decay at least as fast as t−(n+2)/4, no matter how slow the decay of ∥u(·,t)∥L2(Rn) might be.

A Note on Stokes Approximations to Leray Solutions of the Incompressible Navier-Stokes Equations in Rn

In the early 1980s it was well established that Leray solutions of the unforced Navier-Stokes equations in Rn decay in energy norm for large time. With the works of T. Miyakawa, M. Schonbek and others it is now known that the energy decay rate cannot in general be any faster than t^-(n+2)/4 and is typically much slower. In contrast, we show in this note that, given an arbitrary Leray solution u(.,t), the difference of any two Stokes approximations to the Navier-Stokes flow u(.,t) will always decay at least as fast as t^-(n+2)/4, no matter how slow the decay of || u(.,t) ||_L2 might happen to be.

Primal-dual gap estimators for a posteriori error analysis of nonsmooth minimization problems

The primal-dual gap is a natural upper bound for the energy error and, for uniformly convex minimization problems, also for the error in the energy norm. This feature can be used to construct reliable primal-dual gap error estimators for which the constant in the reliability estimate equals one for the energy error and equals the uniform convexity constant for the error in the energy norm. In particular, it defines a reliable upper bound for any functions that are feasible for the primal and the associated dual problem. The abstract a posteriori error estimate based on the primal-dual gap is provided in this article, and the abstract theory is applied to the nonlinear Laplace problem and the Rudin–Osher–Fatemi image denoising problem. The discretization of the primal and dual problems with conforming, low-order finite element spaces is addressed. The primal-dual gap error estimator is used to define an adaptive finite element scheme and numerical experiments are presented, which illustrate the accurate, local mesh refinement in a neighborhood of the singularities, the reliability of the primal-dual gap error estimator and the moderate overestimation of the error.

Convergence Analysis of a Discontinuous Finite Volume Method for the Signorini Problem

We introduce and analyze a discontinuous finite volume method for the Signorini problem. Under suitable regularity assumptions on the exact solution, we derive an optimal a priori error estimate in the energy norm.

Frequency-domain least-squares generalized internal multiple imaging with the energy norm

Recorded seismic data contain various types of scattered energy, including those corresponding to multiples. Traditional imaging techniques are focused on single-scattering events and, thus, may fail to image crucial structures, such as salt flanks and faults that sometimes are illuminated only by the multiple scattered energy. The recently introduced generalized internal multiple imaging (GIMI) method offers an opportunity to image multiples by projecting the recorded data back into the subsurface, followed by an interferometric crosscorrelation of the subsurface wavefield with the recorded data. During this process, the interferometric step converts the first-order scattering to a tomographic component and the double-scattering forms the primary reflectivity. Dealing with a large volume of data consisting of full wavefields over the image space renders the interferometric step computationally expensive in the time domain. To make the implementation of GIMI tractable, we formulate its frequency-domain version. Moreover, we use the energy norm imaging condition to separate the reflectivity portion from the tomographic component. We demonstrate these features with numerical experiments.

Guaranteed a posteriori error bounds for low-rank tensor approximate solutions

We propose a guaranteed and fully computable upper bound on the energy norm of the error in low-rank tensor train (TT) approximate solutions of (possibly) high-dimensional reaction–diffusion problems. The error bound is obtained from Euler–Lagrange equations for a complementary flux reconstruction problem, which are solved in the low-rank TT representation using the block alternating linear scheme. This bound is guaranteed to be above the energy norm of the total error, including the discretization error, the tensor approximation error and the error in the solver of linear algebraic equations, although quadrature errors, in general, can pollute its evaluation. Numerical examples with the Poisson equation and the Schrödinger equation with the Henon–Heiles potential in up to 40 dimensions are presented to illustrate the efficiency of this approach.