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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 the present paper we consider a 1D Poisson model characterized by the presence of an interface, where a transmission condition arises due to jumps of the coefficients. We aim at studying finite element methods with meshes not fitting such an interface. It is well known that when the mesh does not fit the material discontinuities the resulting scheme provides in general lower order accurate solutions. We focus on so-called embedded approaches, frequently adopted to treat fluid–structure interaction problems, with the aim of recovering higher order of approximation also in presence of non fitting meshes; we implement several methods inspired by: the Immersed Boundary method, the Fictitious Domain method, and the Extended Finite Element method. In particular, we present four formulations in a comprehensive and unified format, proposing several numerical tests and discussing their performance. Moreover, we point out issues that may be encountered in the generalization to higher dimensions and we comment on possible solutions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we analyze the standard piece-wise bilinear finite element approximation of a model reaction–diffusion problem. We prove supercloseness results when appropriate graded meshes are used. The meshes are those introduced in Durán and Lombardi (2005) [8] but with a stronger restriction on the graduation parameter. As a consequence we obtain almost optimal error estimates in the -norm thus completing the error analysis given in Durán and Lombardi (2005) [8]. [Copyright &y& Elsevier]

Abstract: Many rate-independent models for metals utilize the gradient of effective plastic strain to capture size-dependent behavior. This enhancement, sometimes termed as “explicit” gradient formulation, requires higher-order tractions to be imposed on the evolving elasto-plastic boundary and the resulting numerical framework is complicated. An “implicit” scalar gradient model was thus developed in Peerlings [Peerlings, R.H.J., 2007. On the role of moving elastic–plastic boundaries in strain gradient plasticity. Model. Simul. Mater. Sci. Eng. 15, 109–120] that has only C0 continuity requirements and its implementation is straightforward. However, both explicit and implicit scalar gradient models can be problematic when the effective plastic strains do not have smooth profiles. To address this limitation, an implicit tensorial gradient model is proposed in this paper based on the generalized micromorphic framework. It is also demonstrated that the scalar and tensorial implicit gradient models give similar results when the effective plastic strains fluctuate smoothly. [Copyright &y& Elsevier]

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]

Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a 'kinetic under-cooling' regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse-interface approximation of the model. Finite-element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions. [ABSTRACT FROM AUTHOR]

Abstract. We consider a fully practical finite element approximation of the nonlinear degenerate parabolic system γ ∂u/∂t - ∇. (b(u)∇[w+αφ]) = 0, w = -γΔu + γ-1 ψ (u), ∇.(c(u)∇φ) = 0 subject to an initial condition u0(·) ∈ [-1, 1] on u and flux boundary conditions on all three equations. Here γ ∈ R>0, α ∈ R≥0, ψ is a nonsmooth double well potential, and c(u) := 1+u, b(u) := 1-u² are degenerate coefficients. The degeneracy in b restricts u(·,·) ∈ [-1, 1]. The above, in the limit γ → 0, models the evolution of voids by surface diffusion in an electrically conducting solid. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions. Furthermore, an iterative scheme for solving the resulting nonlinear discrete system is introduced and analyzed. Finally, some numerical experiments are presented. [ABSTRACT FROM AUTHOR]