522 research outputs found
Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity
In this work we propose and analyze a numerical method for electrical
impedance tomography of recovering a piecewise constant conductivity from
boundary voltage measurements. It is based on standard Tikhonov regularization
with a Modica-Mortola penalty functional and adaptive mesh refinement using
suitable a posteriori error estimators of residual type that involve the state,
adjoint and variational inequality in the necessary optimality condition and a
separate marking strategy. We prove the convergence of the adaptive algorithm
in the following sense: the sequence of discrete solutions contains a
subsequence convergent to a solution of the continuous necessary optimality
system. Several numerical examples are presented to illustrate the convergence
behavior of the algorithm.Comment: 26 pages, 12 figure
A local hybrid surrogate-based finite element tearing interconnecting dual-primal method for nonsmooth random partial differential equations
A domain decomposition approach for high-dimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dual-primal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problem-dependent hp-refinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the KarhunenâLoève expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions. Š 2020 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd
A local hybrid surrogateâbased finite element tearing interconnecting dualâprimal method for nonsmooth random partial differential equations
A domain decomposition approach for highâdimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dualâprimal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problemâdependent hpârefinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the KarhunenâLoève expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions
A stable and optimally convergent LaTIn-CutFEM algorithm for multiple unilateral contact problems
In this paper, we propose a novel unfitted finite element method for the simulation of multiple body contact. The computational mesh is generated independently of the geometry of the interacting solids, which can be arbitrarily complex. The key novelty of the approach is the combination of elements of the CutFEM technology, namely the enrichment of the solution field via the definition of overlapping fictitious domains with a dedicated penalty-type regularisation of discrete operators, and the LaTIn hybrid-mixed formulation of complex interface conditions. Furthermore, the novel P1-P1 discretisation scheme that we propose for the unfitted LaTIn solver is shown to be stable, robust and optimally convergent with mesh refinement. Finally, the paper introduces a high-performance 3D level-set/CutFEM framework for the versatile and robust solution of contact problems involving multiple bodies of complex geometries, with more than two bodies interacting at a single point
Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms
An abstract framework for constructing stable decompositions of the spaces corresponding
to general symmetric positive definite problems into âlocalâ subspaces and a global
âcoarseâ space is developed. Particular applications of this abstract framework include
practically important problems in porous media applications such as: the scalar elliptic
(pressure) equation and the stream function formulation of its mixed form, Stokesâ and
Brinkmanâs equations. The constant in the corresponding abstract energy estimate is shown
to be robust with respect to mesh parameters as well as the contrast, which is defined as
the ratio of high and low values of the conductivity (or permeability). The derived stable
decomposition allows to construct additive overlapping Schwarz iterative methods with
condition numbers uniformly bounded with respect to the contrast and mesh parameters. The
coarse spaces are obtained by patching together the eigenfunctions corresponding to the
smallest eigenvalues of certain local problems. A detailed analysis of the abstract
setting is provided. The proposed decomposition builds on a method of Galvis and Efendiev
[Multiscale Model. Simul. 8 (2010) 1461â1483] developed
for second order scalar elliptic problems with high contrast. Applications to the finite
element discretizations of the second order elliptic problem in Galerkin and mixed
formulation, the Stokes equations, and Brinkmanâs problem are presented. A number of
numerical experiments for these problems in two spatial dimensions are provided
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