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• Guaranteed energy error bounds for the Poisson equation using a flux-free approach: solving the local problems in subdomains

  N. Parés, H. Santos, P. Diez

  Int. J. Numer. Meth. Engng. (2009). Vol. 79 (10), pp. 1203-1244

  
  

  Abstract

  A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain based a posteriori error estimate which yields self-equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux-free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds.

  Abstract
  A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed [...]

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• Exact bounds for linear outputs of the advection-diffusion-reaction equation using flux-free error estimates

  N. Parés, P. Diez, A. Huerta

  Journal on Scientific Computing (SIAM) (2009). Vol. 31 (4), pp. 3064-3089

  
  

  Abstract

  The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The proposed approach is an alternative to the standard residual type estimators (hybrid-flux), circumventing the need of flux-equilibration following a fluxfree error estimation strategy. The presented estimator provides sharper estimates than the ones provided by both the standard hybrid-flux techniques and other flux-free techniques.

  Abstract
  The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. [...]

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• Hierarchical X-FEM for n-phase flow (n>2)

  S. Zlotnik, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (30-32), pp. 2329-2338

  
  

  Abstract

  The eXtended Finite Element Method (X-FEM) has been successfully used in two-phase flow problems involving a moving interface. In order to simulate problems involving more than two phases, the X-FEM has to be further eXtended. The proposed approach is presented in the case of a quasi-static Stokes n-phase flow and it is based on using an ordered collection of level set functions to describe the location of the phases. A level set hierarchy allows describing triple junctions avoiding overlapping or “voids” between materials. Moreover, an enriched solution accounting for several simultaneous phases inside one element is proposed. The interpolation functions corresponding to the enriched degrees of freedom require redefining the associated ridge function accounting for all the level sets. The computational implementation of this scheme involves calculating integrals in elements having several materials inside. An adaptive quadrature accounting for the interfaces locations is proposed to accurately compute these integrals. Examples of the hierarchical X-FEM approach are given for a n-phase Stokes problem in 2 and 3 dimensions.

  Abstract
  The eXtended Finite Element Method (X-FEM) has been successfully used in two-phase flow problems involving a moving interface. In order to simulate problems involving more [...]

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• Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method

  R. Cottereau, P. Diez, A. Huerta

  Computational Mechanics (2009). Vol. 44 (4), pp. 533-547

  
  

  Abstract

  We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are strict in the sense that they refer to the difference between the displacement computed on the FE mesh and the exact displacement, solution of the continuous equations, rather than to the difference between the displacements computed on two FE meshes, one coarse and one refined. This method is based on the resolution of a series of local problems on patches of elements and does not require the resolution of a previous problem of flux equilibration, as happens with other methods. The paper concentrates more specifically on linear solid mechanics issues, and on the assessment of the energy norm of the error, seen as a necessary tool for the estimation of the error in arbitrary quantities of interest (linear functional outputs). Applications in both 2D and 3D are presented

  Abstract
  We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are [...]

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• A simple strategy to assess the error in the numerical wave number of the finite element solution of the Helmholtz equation

  L. Steffens, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (15-16), pp. 1389-1400

  
  

  Abstract

  The standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem, based on the linear functional output representing the quantity of interest. For the assessment of the error in the approximation of the wave number for the Helmholtz problem (also referred to as dispersion or pollution error), this strategy cannot be applied. This is because there is no linear extractor producing the wave number from the solution of the acoustic problem. Moreover, in this context, the error assessment paradigm is reverted in the sense that the exact value of the wave number, κ, is known (it is part of the problem data) and the effort produced in the error assessment technique aims at obtaining the numerical wave number, κH, as a postprocess of the numerical solution, uH. The strategy introduced in this paper is based on the ideas used in the a priori analysis. A modified equation corresponding to a modified wave number κm is introduced. Then, the value of κm such that the modified problem better accommodates the numerical solution uH is taken as the estimate of the numerical wave number κH. Thus, both global and local versions of the error estimator are proposed. The obtained estimates of the dispersion error match the a priori predicted dispersion error and, in academical examples, the actual values of the error in the wave number.

  Abstract
  The standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem, based on the linear functional output representing [...]

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• Bounds for quantities of interest and adaptivity in the Element Free Galerkin method

  Y. Vidal, N. Parés, P. Diez, A. Huerta

  Int. J. Numer. Meth. Engng. (2008). Vol. 76 (11), pp. 1782-1818

  
  

  Abstract

  A novel approach for implicit residual-type error estimation in mesh-free methods and an adaptive refinement strategy are presented. This allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods. The adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision.

  Abstract
  A novel approach for implicit residual-type error estimation in mesh-free methods and an adaptive refinement strategy are presented. This allows computing upper and lower [...]

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• Small-scale gravitational instabilities under the oceans: implications for the evolution of oceanic lithosphere and its expression in geophysical observables

  S. Zlotnik, J. Alfonso, P. Diez, M. Fernández

  Philosophical Magazine (2008). Vol. 88 (28-29), pp. 3197-3217

  
  

  Abstract

  Sublithospheric small-scale convection (SSC) is thought to be responsible for the flattening of the seafloor depth and surface heat flow observed in mature plates. Although the existence of SSC is generally accepted, its ability to effectively produce a constant lithospheric thickness (i.e. flattening of observables) is a matter of debate. Here we study the development and evolution of SSC with a 2D thermomechanical finite-element code. Emphasis is put on (i) the influence of various rheological and thermophysical parameters on SSC, and (ii) its ability to reproduce geophysical observables (i.e. seafloor depth, surface heat flow, and seismic velocities). We find that shear heating plays no significant role either in the onset of SSC or in reducing the lithospheric thickness. In contrast, radiogenic heat sources and adiabatic heating exert a major control on both the vigour of SSC and the thermal structure of the lithosphere. We find that either dislocation creep, diffusion creep, or a combination of these mechanisms, can generate SSC with rheological parameters given by laboratory experiments. However, vigorous SSC and significant lithospheric erosion are only possible for relatively low activation energies.

  Abstract
  Sublithospheric small-scale convection (SSC) is thought to be responsible for the flattening of the seafloor depth and surface heat flow observed in mature plates. Although [...]

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• Modelling gravitational instabilities: Slab break–off and Rayleigh–Taylor diapirism

  S. Zlotnik, M. Fernández, P. Diez, J. Vergés

  Pure and Applied Geophysics (2008). Vol. 165 (8), pp. 1491-1510

  
  

  Abstract

  A non-standard new code to solve multiphase viscous thermo–mechanical problems applied to geophysics is presented. Two numerical methodologies employed in the code are described: A level set technique to track the position of the materials and an enrichment of the solution to allow the strain rate to be discontinuous across the interface. These techniques have low computational cost and can be used in standard desktop PCs. Examples of phase tracking with level set are presented in two and three dimensions to study slab detachment in subduction processes and Rayleigh–Taylor instabilities, respectively. The modelling of slab detachment processes includes realistic rheology with viscosity depending on temperature, pressure and strain rate; shear and adiabatic heating mechanisms; density including mineral phase changes and varying thermal conductivity. Detachment models show a first prolonged period of thermal diffusion until a fast necking of the subducting slab results in the break–off. The influence of several numerical and physical parameters on the detachment process is analyzed: The shear heating exerts a major influence accelerating the detachment process, reducing the onset time to one half and lubricating the sinking of the detached slab. The adiabatic heating term acts as a thermal stabilizer. If the mantle temperature follows an adiabatic gradient, neglecting this heating term must be included, otherwise all temperature contrasts are overestimated. As expected, the phase change at 410 km depth (olivine–spinel transition) facilitates the detachment process due to the increase in negative buoyancy. Finally, simple plume simulations are used to show how the presented numerical methodologies can be extended to three dimensions.

  Abstract
  A non-standard new code to solve multiphase viscous thermo–mechanical problems applied to geophysics is presented. Two numerical methodologies employed in the code are [...]

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• Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the Discontinuous Galerkin time discretization

  N. Parés, P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (19-20), pp. 1641-1660

  
  

  Abstract

  Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh. Exact bounds, that is bounds guaranteed with respect to the exact solution, are needed to properly certify the accuracy of the results, especially if the meshes are coarse. The paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems. In this first part, the bounds account for the error associated with the spatial discretization. The error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error. The time discretization is performed using the discontinuous Galerkin method, both for the primal and adjoint problems. In the error estimation procedure, equilibrated fluxes at interelement edges are calculated using hybridization techniques.

  Abstract
  Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds [...]

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• Bounds of functional outputs for parabolic problems. Part II: Bounds of the exact solution

  N. Parés, P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (19-20), 1661-1679

  
  

  Abstract

  The paper introduces a methodology to compute upper and lower bounds for linear-functional outputs of the exact solutions of parabolic problems. In this second part, the bounds account for the error both in space and time. The assumption stating that the error introduced by the time marching scheme is negligible, used in the first part, is removed here. The bounds are computed starting from an approximation of the exact solution, associated with a spatial mesh and a time grid. Nevertheless, the bounds are guaranteed with respect to the exact solution, with no reference to any mesh or time discretization.

  Abstract
  The paper introduces a methodology to compute upper and lower bounds for linear-functional outputs of the exact solutions of parabolic problems. In this second part, the bounds [...]

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