The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled employing the fixed-stress split scheme, which leads to a semi-discrete system solved iteratively. The error bounds are derived by combining a posteriori estimates for contractive mappings with those of the functional type for elliptic partial differential equations. The estimates are applicable for any approximation in the admissible functional space and are independent of the discretization method. They are fully computable, do not contain mesh dependent constants, and provide reliable global estimates of the error measured in the energy norm. Moreover, they suggest efficient error indicators for the distribution of local errors, which can be used in adaptive procedures.
A posteriori error estimates for approximate solutions of Barenblatt-Biot poroelastic model
University of Bergen（卑尔根大学）；Bergen University College（卑尔根理工学院）；V.A. Steklov Institute of Mathematics in St.-Petersburg（圣彼得堡V.A.斯捷克洛夫数学研究所）；University of Iceland（冰岛大学）
A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the H1(Ω)-conforming Taylor-Hood finite element combination, consisting of polynomial degrees k + 1 for the displacements and the fluid pressure and k for the total pressure. An a posteriori error estimator is derived on the basis of H(div)-conforming reconstructions of the stress and flux approximations. The symmetry of the reconstructed stress is allowed to be satisfied only weakl…
The Biot problem of poroelasticity is extended by Signorini contact conditions. The resulting Biot contact problem is formulated and analyzed as a two field variational inequality problem of a perturbed saddle point structure. We present an a priori error analysis for a general as well as for a $hp$-FE discretization including convergence and guaranteed convergence rates for the latter. Moreover, we derive a family of reliable and efficient residual based a posteriori error estimators, and elaborate how a simple and efficient primal-dual active set solver can be applied to solve the discrete G…
We establish upper bounds of bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs. Here we continue the research started in in our revious publications where computability, in the rigorous sense of computable analysis, has been established for solution operators of Cauchy and dissipative boundary-value problems for such systems.
For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the $L^2$ norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of the conforming finite element method, and has an optimal speed of convergence for the eigenfunctions with the worst regularity. The resulting error estimate bounds the distance of spaces of exact and approximate eigenfunctions and, hence, is robust even in the case of multiple and tightly clustered eigenvalues. The accuracy of the proposed bound is illustrated…
We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed finite element problems. We show that the estimator is reliable up to a prefactor that tends to one with mesh refinement or with polynomial degree increase. We also derive a fully computable upper bound on the prefactor for several common settings of domains and boundary conditions. This leads to a guaranteed estimate without any assumption on the mesh size or th…
This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surround…
Nowadays, a posteriori error control methods have formed a new important part of the numerical analysis. Their purpose is to obtain computable error estimates in various norms and error indicators that show distributions of global and local errors of a particular numerical solution. In this paper, we focus on a particular class of domain decomposition methods (DDM), which are among the most efficient numerical methods for solving PDEs. We adapt functional type a posteriori error estimates and construct a special form of error majorant which allows efficient error control of approximations comp…
Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation
University of Duisburg-Essen（杜伊斯堡-埃森大学）；RICAM, Austrian Academy of Sciences（奥地利科学院RICAM研究所）；University of Jyväskylä（于韦斯屈莱大学）；V. A.˜Steklov Institute of Mathematics in St.˜Petersburg（圣彼得堡V.A.斯捷洛夫数学研究所）
Functional a posteriori error estimates for parabolic time-periodic boundary value problems
Johannes Kepler University Linz（林茨约翰·开普勒大学）；V. A. Steklov Institute of Mathematics in St. Petersburg（圣彼得堡V.A.斯捷克洛夫数学研究所）；University of Jyväskylä（于韦斯屈莱大学）；Johann Radon Institute for Computational and Applied Mathematics（约翰·拉顿计算与应用数学研究所）
We present a local construction of H(curl)-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best-approximations over the entire local versions of H(curl). This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of Pra…
A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems
ENS Cachan/CNRS/Université Paris-Saclay（巴黎高等师范学校卡尚分校/CNRS/巴黎萨克雷大学）；Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS（巴黎东大学，多尺度建模与模拟实验室，MSME UMR 8208 CNRS）
We introduce two a posteriori error estimators for Nédélec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager-Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings,…
Functional a posteriori error estimates for time-periodic parabolic optimal control problems
Johannes Kepler University Linz（林茨约翰·开普勒大学）；V. A. Steklov Institute of Mathematics in St. Petersburg（圣彼得堡V.A.斯捷洛夫数学研究所）；University of Jyväskylä（于韦斯屈莱大学）；Johann Radon Institute for Computational and Applied Mathematics（约翰·拉登计算与应用数学研究所）
Conforming and Non-Conforming Functional A Posteriori Error Estimates for Elliptic Boundary Value Problems in Exterior Domains: Theory and Numerical Tests
Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems
Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution ha…
A posteriori error estimates and adaptivity for locally conservative methods. Inexpensive implementation and evaluation, polytopal meshes, iterative linearization and algebraic solvers, and applications to complex porous media flows
Robust error bounds for finite element approximation of reaction-diffusion problems with non-constant reaction coefficient in arbitrary space dimension
Brown University（布朗大学）；University of Oxford（牛津大学）；Institute of Mathematics, Academy of Sciences（科学院数学研究所）
A posteriori error estimation for coarse or distorted meshes
Universidad de la República（乌拉圭共和国大学）；Université de Strasbourg（斯特拉斯堡大学）；Inria（法国国家数字研究院）；Universidad Técnica Federico Santa María（费德里科·圣玛丽亚技术大学）；Univ. Lille（里尔大学）；Arts et Métiers Institute of Technology（巴黎高科艺术工艺学院）；Centrale Lille（里尔中央理工学院）；Junia（朱尼亚应用科学联盟）
