Local Neumann Problems Research Articles

A semi‐hybrid‐mixed method for Stokes–Brinkman–Darcy flows with H(div)‐velocity fields

SummaryWe consider new semi‐hybrid‐mixed finite element formulations for Stokes–Brinkman problems. Using ‐conforming approximate velocity fields, the continuity of normal components over element interfaces is taken for granted, and pressure is searched in discontinuous spaces preserving the divergence compatibility property. Tangential continuity is weakly imposed by a Lagrange multiplier playing the role of tangential traction. The method is strongly mass‐conservative, leading to exact divergence‐free simulations of incompressible flows. The Lagrange multiplier space requires specific choices according to the velocity approximations implemented in each element geometry. In certain cases, classic divergence‐compatible pairs adopted for Darcy flows may require divergence‐free bubble enrichment to enforce tangential continuity in some extent, avoiding any extra stabilization technique. Considerable improvement in computational performance is achieved by the application of static condensation: the global system is solved only for a piecewise constant pressure variable, velocity normal trace and tangential traction over interfaces. The remaining solution components are recovered by solving independent local Neumann problems in each element. Numerical results are presented for a set of standard test cases in the field of Stokes–Brinkman–Darcy flows with known analytical solutions. The main convergence properties of the method are verified in the whole range of parameters, from Stokes to Darcy limits, as well as for the combined Stokes–Darcy scenario. The robustness of the method is also demonstrated for more challenging test problems with complex non‐conforming meshes with local refinement patterns as well as for curved geometry.

Revisiting the robustness of the multiscale hybrid-mixed method: The face-based strategy

This work proposes a new finite element for the multiscale hybrid-mixed method (MHM) applied to the Poisson equation with highly oscillatory coefficients. Unlike the original MHM method, multiscale bases are the solution to local Neumann problems driven by piecewise continuous polynomial interpolation on the skeleton faces of the macroscale mesh. As a result, we prove the optimal convergence of MHM by refining the face partition and leaving the mesh of macroelements fixed. This property allows the MHM method to be resonance free under the usual assumptions of local regularity. The numerical analysis of the method also revisits and complements the original approach proposed by D. Paredes, F. Valentin and H. Versieux (2017). Numerical experiments assess the new theoretical results.

Hierarchical a posteriori error estimation of Bank–Weiser type in the FEniCS Project

In the seminal paper of Bank and Weiser (1985) [17] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank–Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel algorithmic approach to constructing hierarchical estimators of the Bank–Weiser type that is designed for implementation in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement for diverse Poisson problems and for mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. Two open source implementations in the DOLFIN and DOLFINx solvers of the FEniCS Project are provided as supplementary material.

The MHM Method for Linear Elasticity on Polytopal Meshes

Abstract The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn’s inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case.

A multiscale mixed finite element method applied to the simulation of two-phase flows

The multiscale hybrid mixed finite element method (MHM-H(div)), previously developed for Darcy’s problems, is extended for coupled flow/pressure and transport system of two-phase flow equations on heterogeneous media under the effect of gravitational segregation. It is combined with an implicit transport solver in a sequential fully implicit (SFI) manner. The MHM-H(div) method is designed to cope with the complex geometry and inherent multiscale nature of the phenomena. The discretizations are based on a general domain partition formed by polyhedral subregions, where a hierarchy of meshes and approximation spaces are considered. The multiscale approach is applied to the flux/pressure kernel making use of coarse scale normal fluxes between subregions (trace variable). The fine-scale features inside each subregion are determined by resolving completely independent local Neumann problems, the boundary conditions being set by the trace variable, by the mixed finite element method using fine flux and pressure representations. These properties imply that the MHM-H(div) can be interpreted as a classical mixed formulation of the model problem in the whole domain, based on a H(div)-conforming space with normal components over the macro-partition interfaces constrained by the trace space, and showing divergence compatibility with the pressure space. Consequently, local mass conservation is observed at the micro-scale elements inside the subregions, an essential property for flows in heterogeneous media, and divergence-free constraint strongly enforced for incompressible flows. The efficient use of static condensation leads to a global system to be solved only in terms of primary degrees of freedom associated with the trace variable and of a piecewise constant pressure for each subregion. This procedure allows a substantial reduction of the dominant computational costs associated with the flux/pressure kernel embedded in the numerical model. An iterative coupling technique is adopted to solve the two-phase flow equations using a shared integration point memory implementation model. At each SFI time step, the efficiency iterative method for the transport equations is improved using a Quasi-Newton method with a simple but effective nonlinear acceleration. The numerical examples show that the proposed scheme is able to solve challenging coupled flow and transport problems.

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

Guaranteed A Posteriori Error Estimates for a Staggered Discontinuous Galerkin Method

In this paper, we present for the first time guaranteed upper bounds for the staggered discontinuous Galerkin method for diffusion problems. Two error estimators are proposed for arbitrary polynomial degrees and provide an upper bound on the energy error of the scalar unknown and $$L^2$$ -error of the flux, respectively. Both error estimators are based on the potential and flux reconstructions. The potential reconstruction is given by a simple averaging operator. The equilibrated flux reconstruction can be found by solving local Neumann problems on elements sharing an edge with a Raviart–Thomas mixed method. Reliability and efficiency of the two a posteriori error estimators are proved. Numerical results are presented to validate the theoretical results.

Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis

We derive equilibrated reconstructions of the Darcy velocity and of the total stress tensor for Biot’s poro-elasticity problem. Both reconstructions are obtained from mixed finite element solutions of local Neumann problems posed over patches of elements around mesh vertices. The Darcy velocity is reconstructed using Raviart–Thomas finite elements and the stress tensor using Arnold–Winther finite elements so that the reconstructed stress tensor is symmetric. Both reconstructions have continuous normal component across mesh interfaces. Using these reconstructions, we derive a posteriori error estimators for Biot’s poro-elasticity problem, and we devise an adaptive space–time algorithm driven by these estimators. The algorithm is illustrated on test cases with analytical solution, on the quarter five-spot problem, and on an industrial test case simulating the excavation of two galleries.

Balancing Domain Decomposition by Constraints and Perturbation

In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC has the same polylogarithmic bound for the condition number as the standard formulation. Two types of properly scaled zero-order perturbations are considered: one uses a mass matrix, and the other uses a Robin-type boundary condition, i.e, a mass matrix on the interface. With perturbation, the wellposedness of the local Neumann problems and the global coarse problem is automatically guaranteed, and coarse degrees of freedom can be defined only for convergence purposes but not well-posedness. This allows a much simpler implementation as no complicated corner selection algorithm is needed. Minimal coarse spaces using only face or edge constraints can also be considered. They are very useful in extreme scale calculations where the coarse problem is usually the bottleneck that can jeopardize scalability. The perturbation also adds extra robustness as the perturbed formulation works even when the constraints fail to eliminate a small number of subdomain rigid body modes from the standard BDDC space. This is extremely important when solving problems on unstructured meshes partitioned by automatic graph partitioners since arbitrary disconnected subdomains are possible. Numerical results are provided to support the theoretical findings.

Cell Conservative Flux Recovery and A Posteriori Error Estimate of Vertex-Centered Finite Volume Methods

A cell conservative flux recovery technique is developed here for vertexcentered finite volume methods of second order elliptic equations. It is based on solving a local Neumann problem on each control volume using mixed finite element methods. The recovered flux is used to construct a constant free a posteriori error estimator which is proven to be reliable and efficient. Some numerical tests are presented to confirm the theoretical results. Our method works for general order finite volume methods and the recovery-based and residual-based a posteriori error estimators is the first result on a posteriori error estimators for high order finite volume methods. AMS subject classifications: 65N15, 65N30, 65N50

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.

A posteriori error estimation based on conservative flux reconstruction for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem on anisotropic meshes

Based on conservative flux reconstruction, we derive a posteriori error estimates for the nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem on anisotropic meshes, since the solution exhibits boundary or interior layers and this anisotropy is reflected in a discretization by using anisotropic meshes. Without the assumption that the meshes are shape-regular, our estimates give the upper bounds on the error only containing the alignment measure factor, and therefore, could provide actual numerical bounds if the alignment measure was approximated very well. Two resulting error estimators are presented to be equivalent to each other up to a data oscillation, one of which can be directly constructed without solving local Neumann problems and provide computable error bound. Numerical experiments confirm that our estimates are reliable and efficient as long as the singularly perturbed problem is discretized by a suitable mesh which leads to a small alignment measure.

Cell Conservative Flux Recovery and A Posteriori Error Estimate of Vertex-Centered Finite Volume Methods

AbstractA cell conservative flux recovery technique is developed here for vertex-centered finite volume methods of second order elliptic equations. It is based on solving a local Neumann problem on each control volume using mixed finite element methods. The recovered flux is used to construct a constant freea posteriorierror estimator which is proven to be reliable and efficient. Some numerical tests are presented to confirm the theoretical results. Our method works for general order finite volume methods and the recovery-based and residual-baseda posteriorierror estimators is the first result ona posteriorierror estimators for high order finite volume methods.

On a posteriori error estimates for the linear triangular finite element

Based on equilibration of side fluxes, an a posteriori error estimator is obtained for the linear triangular element for the Poisson equation, which can be computed locally. We present a procedure for constructing the estimator in which we use the Lagrange multiplier similar to the usual equilibrated residual method introduced by Ainsworth and Oden. The estimator is shown to provide guaranteed upper bound, and local lower bounds on the error up to a multiplicative constant depending only on the geometry. Based on this, we give another error estimator which can be directly constructed without solving local Neumann problems and also provide the two-sided bounds on the error. Finally, numerical tests show our error estimators are very efficient.

Four closely related equilibrated flux reconstructions for nonconforming finite elements

We consider the Crouzeix–Raviart nonconforming finite element method for the Laplace equation. We present four equilibrated flux reconstructions, by direct prescription or by mixed approximation of local Neumann problems, either relying on the original simplicial mesh only or employing a dual mesh. We show that all these reconstructions coincide provided the underlying system of linear algebraic equations is solved exactly. We finally consider an inexact algebraic solve, adjust the flux reconstructions, and point out the differences.

Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids

Discontinuous Galerkin methods handle very well general polygonal and nonmatching meshes. We present in this Note a H ( div ) -conforming reconstruction of the flux on such meshes in the setting of an elliptic problem. We exploit the local conservation property of discontinuous Galerkin methods and solve local Neumann problems by means of the Raviart–Thomas–Nédélec mixed finite element method. Our reconstruction can be used in a guaranteed a posteriori error estimate and it is also of independent interest when the approximate flux is to be used subsequently in a transport problem. To cite this article: A. Ern, M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity

In our earlier work [E. Stein, S. Ohnimus, Coupled model- and solution-adaptivity in the finite-element method, Comput. Methods Appl. Mech. Engrg. 150 (1997) 327–350; E. Stein, S. Ohnimus, Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems, Comput. Methods Appl. Mech. Engrg. 176 (1999) 363–385; E. Stein, M. Rüter, S. Ohnimus, Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends, Int. J. Numer. Methods Engrg. 60 (2004) 103–138.], residual-based combined model and finite element discretization error estimators as a basis for adaptivity were derived that rely on the solutions of Neumann problems on the element level. In particular, these estimators control the errors obtained while expanding the model hierarchically from 1D- to 2D- and from 2D- to 3D-elastic theories in the (global) energy norm. In this paper, we extend both types of estimates to goal-oriented a posteriori error estimation within the framework of nonlinear structural mechanics. Structural model adaptivity is characterized by the fact that the dimensions, in general the constitutive equations and especially the related subdomains—and thus the test and solution spaces of the process-dependent models considered—are different. This has the important consequence that consistent prolongations of kinematic and static nodal quantities of finite elements of the coarser model “ m” to those of the refined model “ m + 1” are necessary. Therefore, such prolongation operators are discussed in detail. Furthermore, duality techniques are used for the model and discretization error estimates, using system and load-dependent quantities of interest. In this paper, we also consider different finite element meshes for the primal and the dual problem. Finally, some numerical examples demonstrate the expected effects, realizing verification and even validation in structural mechanics, i.e. approaching a new paradigm in Computational Mechanics for achieving reliability and efficiency.

A posteriori error estimators for locally conservative methods of nonlinear elliptic problems

In this article we present a unified analysis of some implicit and explicit error estimators for a general class of locally conservative methods applied to nonlinear elliptic problems. The implicit error estimator is computed by solving local Neumann problems, like the Bank–Weiser type, where the Neumann data are provided in a natural way by the locally conservative methods. This can be considered as the equilibrated residual method extended to locally conservative methods which are typically based on discontinuous approximation spaces. We also derive some explicit error estimators of residual type by bounding the implicit error estimator further from above. Our analysis is based on a proper decomposition of the error itself, and seems more direct than other approaches based on the Helmholtz decomposition of the gradient of the error.

Neumann–Neumann–Schur complement methods for Fekete spectral elements

For the iterative solution of the Schur complement system associated with the discretization of an elliptic problem by means of a triangular spectral element method (TSEM), Neumann-Neumann (NN) type preconditioners are constructed and studied. The TSEM approximation, based on Fekete nodes, is a generalization to non-tensorial elements of the classical Gauss-Lobatto-Legendre quadrilateral spec- tral elements. Numerical experiments show that the TSEM Schur complement condition number grows linearly with the polynomial approximation degree, N, and quadratically with the inverse of the mesh size, h. NN preconditioners for the Schur complement allow to reduce the N-dependence of the condition number, by solving local Neumann problems on each spectral element, and to eliminate the h-dependence if an additional coarse solver is employed. Numerical results indicate that, in spite of the more severe ill-conditioning, the condition number of the TSEM preconditioned operator satisfies the same bound as that of the standard SEM, i.e., Ch −2 (1 + logN) 2 for one-level NN preconditioning and C(1 + logN) 2 for two-level Balancing Neumann-Neumann (BNN) preconditioning.