In this paper, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto finite elements in space. A truncated Galerkin orthogonality is used to obtain duality-based a posteriori error estimates that address the temporal and the spatial error separately. Based on this theory, a space-time adaptive solver for the Schrödinger equation is devised. An efficient matrix-free implementation of the differential operator, suited for spectral elements, is used to enable computations for realistic configurations. We demonstrate the performance of the algorithm for the example of matter-field interaction.

We construct and analyze conservative local discontinuous Galerkin (LDG) methods for the Generalized Korteweg-de-Vries equation. LDG methods are designed by writing the equation as a system and performing separate approximations to the spatial derivatives. The main focus is on the development of conservative methods which can preserve discrete versions of the first two invariants of the continuous solution, and a posteriori error estimates for a fully discrete approximation that is based on the idea of dispersive reconstruction. Numerical experiments are provided to verify the theoretical estimates.

A uniaxial perfectly matched layer (PML) method is proposed for solving the scattering problem with multiple cavities. By virtue of the integral representation of the scattering field, we decompose the problem into a system of single-cavity scattering problems which are coupled with Dirichlet-to-Neumann maps. A PML is introduced to truncate the exterior domain of each cavity such that the computational domain does not intersect those for other cavities. Based on the a posteriori error estimates, an adaptive finite element algorithm is proposed to solve the coupled system. The novelty of the proposed method is that its computational complexity is comparable to that for solving uncoupled single-cavity problems. Numerical experiments are presented to demonstrate the efficiency of the adaptive PML method.

A weakly over-penalized symmetric interior penalty method is applied to solve elliptic eigenvalue problems. We derive a posteriori error estimator of residual type, which proves to be both reliable and efficient in the energy norm. Some numerical tests are provided to confirm our theoretical analysis.

In this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to demonstrate the effectiveness of our method.

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions, where the wave propagation is governed by the Helmholtz equation. The scattering problem is modeled as a boundary value problem over a bounded domain. Based on the Dirichlet-to-Neumann (DtN) operator, a transparent boundary condition is introduced on an artificial circular boundary enclosing the obstacle. An adaptive finite element based on a posterior error estimate is presented to solve the boundary value problem with a nonlocal DtN boundary condition. Numerical experiments are included to compare with the perfectly matched layer (PML) method to illustrate the competitive behavior of the proposed adaptive method.

In this paper, hp-adaptive finite element methods are studied for time-harmonic Maxwell’s equations. We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates. Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities, edge singularities, and an engineering benchmark problem of Maxwell’s equations. The hp-adaptive methods show much better performance than the h-adaptive method.

In this paper, we derive a posteriori error estimates for finite element approximations of the optimal control problems governed by the Stokes-Darcy system. We obtain a posteriori error estimators for both the state and the control based on the residual of the finite element approximation. It is proved that the a posteriori error estimate provided in this paper is both reliable and efficient.

We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of functional type. An efficient numerical strategy is based upon balancing the modeling and discretization errors, which provides an economical way of finding an approximate solution with an a priori given accuracy. Numerical tests demonstrate the reliability and efficiency of this combined modeling-discretization method.

We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear)interface problem for the 2D Laplacian. We introduce some new a posteriorierror estimators based on the (h − h/2)-errorestimation strategy. In particular, these include the approximation error for the boundarydata, which allows to work with discrete boundary integral operators only. Using theconcept of estimator reduction, we prove that the proposed adaptive algorithm isconvergent in the sense that it drives the underlying error estimator to zero. Numericalexperiments underline the reliability and efficiency of the considered adaptivemesh-refinement.

In this paper, a new type of gradient recovery method based on vertex-edge-face interpolation is introduced and analyzed. This method gives a new way to recover gradient approximations and has the same simplicity, efficiency, and superconvergence properties as those of superconvergence patch recovery method and polynomial preserving recovery method. Here, we introduce the recovery technique and analyze its superconvergence properties. We also show a simple application in the a posteriori error estimates. Some numerical examples illustrate the effectiveness of this recovery method.

The primary objective of this work is to develop coarse-grainingschemes for stochastic many-body microscopic models and quantify theireffectiveness in terms of a priori and a posteriori error analysis. Inthis paper we focus on stochastic lattice systems ofinteracting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grainedapproximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first – and often inadequate – approximation and obtain more accurate coarse-graining schemes.The cluster expansions yield also sharp a posteriori error estimates forthe coarse-grained approximations that can be used for the construction ofadaptive coarse-graining methods.
We present a number of numerical examples that demonstrate that thecoarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improvepredictions of earlier coarse-graining schemes for short-range interactions.

This work represents a first step towards the simulation of the motion of water in a complex hydrodynamic configuration, such asa channel network or a river delta, by means of a suitable“combination” of different mathematicalmodels. In this framework a wide spectrum of space and time scales is involveddue to the presence of physical phenomena ofdifferent nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solvethroughout the whole domain the most complex model (with solution $u_{\rm{fine}}$ ) to accuratelydescribe all the physical features of the problem at hand. In our approach instead,for a user-defined output functional ${\cal F}$ , we aim to approximate,within a prescribed tolerance τ, the value ${\cal F}(u_{\rm{fine}})$ by means of the quantity ${\cal F}(u_{\rm{adapted}})$ , $u_{\rm{adapted}}$ being the so-called adapted solution solving thesimpler models on most of the computational domain while confining the complex ones only on a restricted region. Moving fromthe simplified setting where only two hydrodynamic models, fine and coarse, are considered, weprovide an efficient tool able to automatically selectthe regions of the domain where the coarsemodel rather than the fine one are to be solved, while guaranteeing $|{\cal F}(u_{\rm{fine}})-{\cal F}(u_{\rm{adapted}})|$ below the tolerance τ.This goal is achieved via a suitable a posteriori modeling error analysisdeveloped in the framework of a goal-oriented theory.We extend the dual-based approach provided in [Braack and Ern,Multiscale Model Sim.1 (2003) 221–238],for steady equations to the case of a generic time-dependent problem.Then this analysis is specialized to the case we areinterested in, i.e. the free-surface flows simulation, by emphasizing the crucial issue of the time discretization for both the primal and the dualproblems. Finally, inthe last part of the paper a widespread numerical validation is carried out.