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Abstract: This is a paper out of a series of three papers devoted to model- and hpq-adaptive finite element methods assigned for modeling and analysis of complex structures. In this paper we focus our attention on hierarchical models and hierarchical approximations, while the issues of error estimation and adaptivity control will be presented in the next papers of the series. In all the papers we consider the elastic structures of complex mechanical description. We apply 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within our finite element formulation. In the proposed approach, the mechanical model and discretization parameters, such as the size h of the element, and the longitudinal and transverse approximation orders, p and q, can vary locally, i.e. they can be different in each finite element. The a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors. The error-controlled adaptive procedures are derived from a four-step strategy, with possible iterations on h- and p-steps. [ABSTRACT FROM AUTHOR]

In this paper we focus on the thermo-mechanical model proposed in Bourdin et al. (2014) and Marigo et al. (2013) which describes crack genesis and propagation in brittle materials induced by a thermal shock. Our goal is to provide an efficient numerical technique which employs a computational finite element mesh finely customized to the problem at hand to simulate such phenomena in a 2D setting. In particular, we generate automatically adapted anisotropic grids able to closely follow the narrow bands of the damage, driven by a theoretically sound mathematical tool. We carry out three numerical tests to assess the computational performance of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider numerical methods for integro-differential problems based on time discretization via Laplace transformation. We focus our attention in models arising in the context of non Fickian solute transport phenomena in porous media. The mathematical models which describe the evolution of the solute concentrations are characterized by Volterra equations. We present and analyze an hybrid method which combines the Laplace transformation with respect to the time variable with the finite element discretization in the spatial variables. Numerical results illustrate the performance of the method. [Copyright &y& Elsevier]

Abstract: The shallow water equations are used to model large-scale surface flow in the ocean, coastal rivers, estuaries, salt marshes, bays, and channels. They can describe tidal flows as well as storm surges associated with extreme storm events, such as hurricanes. The resulting currents can transport bed load and suspended sediment and result in morphological changes to the seabed. Modeling these processes requires tightly coupling a bed morphology equation to the shallow water equations. Discontinuous Galerkin finite element methods are a natural choice for modeling this coupled system, given the need to solve these problems on unstructured computational meshes, as well as the desire to implement hp-adaptivity for capturing the dynamic features of the solution. However, because of the presence of non-conservative products in the momentum equations, the standard DG method cannot be applied in a straightforward manner. To rectify this situation, we summarize and follow an extended approach described by Rhebergen et al., which uses theoretical results due to Dal Maso et al. appearing in earlier work. In this paper, we focus on aspects of the implementation of the morphological model for bed evolution within the Advanced Circulation (ADCIRC) modeling framework, as well as the verification of the RKDG method in both h (mesh spacing) and p (polynomial order). This morphological model is applied to a number of coastal engineering problems, and numerical results are presented, with attention paid to the effects of h- and p-refinement in these applications. In particular, it is observed that for sediment transport, piecewise constant (i.e., finite volume) approximations of the bed are very over-diffusive and lead to poor sediment solutions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce an hp finite element method to solve a two-dimensional fluid–structure spectral problem. This problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid. We prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues. We define an a posteriori error estimator of the residual type which can be computed locally from the approximate eigenpair. We show its reliability and efficiency by proving that the estimator is equivalent to the energy norm of the error up to higher order terms, the equivalence constant of the efficiency estimate being suboptimal in that it depends on the polynomial degree. We present an hp adaptive algorithm and several numerical tests which show the performance of the scheme, including some numerical evidence of exponential convergence. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the grad–div stabilization for the incompressible Navier–Stokes finite element approximations is considered from two different viewpoints: (i) as a variational multiscale approach for the pressure subgrid modeling and (ii) as a stabilization procedure of least-square type. Some new error estimates for the linearized problem with the grad–div stabilization are proved with the help of norms induced by the pressure Schur complement operator. We discuss the stabilization parameter choice arising in the frameworks of least-square and multiscale methods and consider assumptions which allow to relate both approaches. [Copyright &y& Elsevier]

Abstract: This paper introduces and analyzes two mathematical models for coupling the incompressible Navier–Stokes equations with the porous media flow equations. A numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium, is proposed. Existence and uniqueness results under small data condition of the numerical solution are proved. Optimal a priori error estimates are derived. Numerical examples comparing the two models under varying physical parameters are provided. [Copyright &y& Elsevier]

Abstract: A new family of C 0 Kirchhoff plate elements has been introduced by the authors in the theoretical counterpart of the present paper; A family of C 0 finite elements for Kirchhoff plates I: Error analysis. The method presented is a displacement formulation with the deflection and the rotation as unknowns. In the theoretical part, an a priori and an a posteriori error analysis has been accomplished for the family. In the present contribution, the authors first give a supplementary derivation of the method and recall the main theoretical results, then focus on the computational aspects of the method, and finally present a set of numerical results on various benchmark computations. These tests verify the optimal convergence rate of the method and illustrate the robustness of the reliable and efficient residual-based a posteriori error estimator for adaptive mesh refinements. [Copyright &y& Elsevier]

In this paper we develop and analyze pressure stabilized, finite element methods for the solution of the transient Stokes problem, which is a linear model problem of transient incompressible flow. A model for the bubble enrichment is proposed to stabilize the numerical solution and an implicit backward Euler method is considered for the time approximation. Different methods are obtained depending on whether bubbles are allowed to evolve with time or they are considered quasi-static. Stability estimates are provided for the two methods considered, both for the velocity and the pressure solutions. Numerical results obtained on test cases are also given, which confirm the theoretical stability results. [Copyright &y& Elsevier]