An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures

Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an unbounded domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.

Extrapolation Method for Non-Linear Weakly Singular Volterra Integral Equation with Time Delay

This paper proposes an extrapolation method to solve a class of non-linear weakly singular kernel Volterra integral equations with vanishing delay. After the existence and uniqueness of the solution to the original equation are proved, we combine an improved trapezoidal quadrature formula with an interpolation technique to obtain an approximate equation, and then we enhance the error accuracy of the approximate solution using the Richardson extrapolation, on the basis of the asymptotic error expansion. Simultaneously, a posteriori error estimate for the method is derived. Some illustrative examples demonstrating the efficiency of the method are given.

Error control for statistical solutions of hyperbolic systems of conservation laws

Statistical solutions have recently been introduced as an alternative solution framework for hyperbolic systems of conservation laws. In this work, we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations obtained from the Runge-Kutta Discontinuous Galerkin method in one spatial dimension, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of the residuals and verify their splitting.

A Posteriori Error Analysis for a Lagrange Multiplier Method for a Stokes/Biot Fluid-Poroelastic Structure Interaction Model

In this work, we develop an a posteriori error analysis of a conforming mixed finite element method for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium on isotropic meshes in ℝ d , d ∈ 2 , 3 . The approach utilizes a Lagrange multiplier method to impose weakly the interface conditions [Ilona Ambartsumyan et al., Numerische Mathematik, 140 (2): 513-553, 2018]. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, Helmholtz decomposition, and local approximation properties of the Clément interpolant. On the other hand, inverse inequalities and the localization technique based on simplexe-bubble and face-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

Adaptive modelling of variably saturated seepage problems

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

Adaptive Goal-Oriented Solver for the Linearized Poisson- Boltzmann Equation

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. The new solver is based on a derivation of a new goal-oriented a posteriori error estimate for the electrostatic coupling. This estimate involves the consideration of the primal and adjoint problems for the electrostatic potential of the system, where the goal functional requires pointwise evaluations of the potential. A common practice to treat such functionals is to regularize them, for example, by averaging over balls or by mollification, and to use weak formulations in standard Sobolev spaces. However, this procedure changes the goal functional and may require the numerical evaluation of integrals of discontinuous functions. We overcome the conceptual shortcomings of this approach by using weak formulations involving nonstandard Sobolev spaces and deriving a representation of the error in the goal quantity which does not require averaging and directly exploits the original goal functional. The accuracy of this solver is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths. All the numerical experiments are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.<br><br>

Adaptive Goal-Oriented Solver for the Linearized Poisson- Boltzmann Equation

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. The new solver is based on a derivation of a new goal-oriented a posteriori error estimate for the electrostatic coupling. This estimate involves the consideration of the primal and adjoint problems for the electrostatic potential of the system, where the goal functional requires pointwise evaluations of the potential. A common practice to treat such functionals is to regularize them, for example, by averaging over balls or by mollification, and to use weak formulations in standard Sobolev spaces. However, this procedure changes the goal functional and may require the numerical evaluation of integrals of discontinuous functions. We overcome the conceptual shortcomings of this approach by using weak formulations involving nonstandard Sobolev spaces and deriving a representation of the error in the goal quantity which does not require averaging and directly exploits the original goal functional. The accuracy of this solver is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths. All the numerical experiments are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.<br><br>

Computation of Nonlinear Thermoelectric Effects with Adaptive Methods

An implementation for a fully automatic adaptive finite element method (AFEM) for computation of nonlinear thermoelectric problems in three dimensions is presented. Adaptivity of the nonlinear solvers is based on the well-established hp-adaptivity where the mesh refinement and the polynomial order of elements are methodically controlled to reduce the discretization errors of the coupled field variables temperature and electric potential. A single mesh is used for both fields and the nonlinear coupling of temperature and electric potential is accounted in the computation of a posteriori error estimate where the residuals are computed element-wise. Mesh refinements are implemented for tetrahedral mesh such that conformity of elements with neighboring elements is preserved. Multiple nonlinear solution steps are assessed including variations of the fixed-point method with Anderson acceleration algorithms. The Barzilai-Borwein algorithm to optimize the nonlinear solution steps are also assessed. Promising results have been observed where all the nonlinear methods show the same accuracy with the tendency of approaching convergence with more elements refining. Anderson acceleration is the most efficient among the nonlinear solvers studied where its total computing time is less than half of the more conventional fixed-point iteration.