Self-Adaptive Methods for PDE

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Goal-oriented error estimation for transient parabolic problems

This work focuses on controlling the error and adapting the discretization in the context of parabolic problems. In order to obtain a sound mathematical framework, the time domain is discretized using a Discontinuous Galerkin (DG) approach. This allows to formulate the time stepping procedure in a variational format. The error is measured in the basis of an output of interest of the solution, defined by a linear functional. A dual problem, associated with this linear output is introduced. The dual problem has to be solved backward in time. An error representation is introduced, based on the weak residual of the primal error applied to the dual solution. Two different alternatives are studied to estimate the error in the dual solution: 1) recovery based error estimators and 2) implicit residual type estimators. Once the error assessment is performed implicitly in the dual problem, the obtained estimate is plugged into the primal residual to obtain the error in the quantity of interest. The implementation of the estimator is drastically simplified by using the weak version of the residual instead of the strong version used in previous works. Thus, the output error is assessed using a mixed technique, explicit for the primal problem and implicit for the dual. In the framework of adaptive computations of transient problems, this approach is very attractive because it allows using first the implicit scheme for the dual problem and then integrating the primal problem, estimating the error explicitly and eventually adapting the space-time grid. Thus, at every time step of the time marching scheme, the estimate of the dual error is injected into the primal residual (explicit estimate for the primal problem)

Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery

[EN] Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local error in quantities of interest (QoI), which are often more pertinent than local errors in energy for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems). These GOEE are one of the key unsolved problems of advanced engineering applications in, for example, the aerospace industry. This work presents a simple recovery-based error estimation technique for QoIs whose main characteristic is the use of an enhanced version of the Superconvergent Patch Recovery (SPR) technique previously used for error estimation in the energy norm. This enhanced SPR technique is used to recover both the primal and dual solutions. It provides a nearly statically admissible stress field that results in accurate estimations of the local contributions to the discretisation error in the QoI and, therefore, in an accurate estimation of this magnitude. This approach leads to a technique with a reasonable computational cost that could easily be implemented into already available finite element codes, or as an independent postprocessing tool.This work was supported by the EPSRC Grant EP/G042705/1 "Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method". Stephane Bordas also thanks partial funding for his time provided by the European Research Council Starting Independent Research Grant (ERC Stg Grant Agreement No. 279578) "RealTCut Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery". This work has received partial support from the research project DPI2010-20542 of the Ministerio de Economia y Competitividad (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politecnica de Valencia and Generalitat Valenciana (PROMETEO/2012/023) are also acknowledged. All authors also thank the partial support of the Framework Programme 7 Initial Training Network Funding under Grant No. 289361 "Integrating Numerical Simulation and Geometric Design Technology."González Estrada, OA.; Nadal Soriano, E.; Ródenas, J.; Kerfriden, P.; Bordas, S.; Fuenmayor Fernández, FJ. (2014). Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Computational Mechanics. 53(5):957-976. https://doi.org/10.1007/s00466-013-0942-8S957976535Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. 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A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

International audienceWe consider some (anisotropic and piecewise constant) diffusion problems in domains of R^2, approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is super convergent. The reliability and efficiency of the proposed estimator in confirmed by some numerical tests

Scalable Recovery-based Adaptation on Quadtree Meshes for Advection-Diffusion-Reaction Problems

We propose a mesh adaptation procedure for Cartesian quadtree meshes, to discretize scalar advection-diffusion-reaction problems. The adaptation process is driven by a recovery-based a posteriori estimator for the $L^2(\Omega)$-norm of the discretization error, based on suitable higher order approximations of both the solution and the associated gradient. In particular, a metric-based approach exploits the information furnished by the estimator to iteratively predict the new adapted mesh. The new mesh adaptation algorithm is successfully assessed on different configurations, and turns out to perform well also when dealing with discontinuities in the data as well as in the presence of internal layers not aligned with the Cartesian directions. A cross-comparison with a standard estimate--mark--refine approach and with other adaptive strategies available in the literature shows the remarkable accuracy and parallel scalability of the proposed approach