Elliptic Diffusion Problems Research Articles

Hp-Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

Singularity extraction for elliptic equations with coefficients with jumps and Dirac sources

We explore the mathematical structure of the solution to an elliptic diffusion problem with point-wise Dirac sources. The conductivity parameter is space-varying, may have jumps and the Dirac sources may be located along the discontinuity curves of that parameter. The variational problem, issued by duality, is proven to be well posed using a sharp elliptic regularity result by Di-Giorgi [Mem. Accad. Sci. Torino, 3, 1957]. The paper is aimed at a key expansion into a split singular/regular contributions. The singular part is calculated by an explicit formula, while the regular correction can be computed as the solution to a standard variational Poisson problem. The latter can be successfully approximated by most of the numerical methods practiced nowadays. Some analytical examples are discussed at last to assess the minimality of the assumptions we use to establish our theoretical results.

Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steered p-Robust Multigrid Solvers

Abstract In this work, we study a local adaptive smoothing algorithm for a-posteriori-steered p-robust multigrid methods. The solver tackles a linear system which is generated by the discretization of a second-order elliptic diffusion problem using conforming finite elements of polynomial order p ≥ 1 {p\geq 1} . After one V-cycle (“full-smoothing” substep) of the solver of [A. Miraçi, J. Papež, and M. Vohralík, A-posteriori-steered p-robust multigrid with optimal step-sizes and adaptive number of smoothing steps, SIAM J. Sci. Comput. 2021, 10.1137/20M1349503], we dispose of a reliable, efficient, and localized estimation of the algebraic error. We use this existing result to develop our new adaptive algorithm: thanks to the information of the estimator and based on a bulk-chasing criterion, cf. [W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 1996, 3, 1106–1124], we mark patches of elements with increased estimated error on all levels. Then, we proceed by a modified and cheaper V-cycle (“adaptive-smoothing” substep), which only applies smoothing in the marked regions. The proposed adaptive multigrid solver picks autonomously and adaptively the optimal step-size per level as in our previous work but also the type of smoothing per level (weighted restricted additive or additive Schwarz) and concentrates smoothing to marked regions with high error. We prove that, under a numerical condition that we verify in the algorithm, each substep (full and adaptive) contracts the error p-robustly, which is confirmed by numerical experiments. Moreover, the proposed algorithm behaves numerically robustly with respect to the number of levels as well as to the diffusion coefficient jump for a uniformly-refined hierarchy of meshes.

A-Posteriori-Steered $p$-Robust Multigrid with Optimal Step-Sizes and Adaptive Number of Smoothing Steps

We develop a multigrid solver steered by an a posteriori estimator of the algebraic error. We adopt the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree $p ≥ 1$. Our solver employs zero pre- and one post-smoothing by the overlapping Schwarz (block-Jacobi) method and features an optimal choice of the step-sizes in the smoothing correction on each level by line search. This leads to a simple Pythagorean formula of the algebraic error in the next step in terms of the current error and level-wise and patch-wise error reductions. We show the two following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree $p$; and the estimator represents a two-sided $p$-robust bound on the algebraic error. The $p$-robustness results are obtained by carefully applying the results of Schoberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1–24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of Xu et al. [Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599–659]. We consider quasi-uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal $H^1$-regularity and complete independence for $H^2$-regularity. We also present a simple and effective way for the solver to adaptively choose the number of post-smoothing steps necessary at each individual level, yielding a yet improved error reduction. Numerical tests confirm $p$-robustness and show the benefits of the adaptive number of smoothing steps.

A quasi-optimal variant of the hybrid high-order method for elliptic partial differential equations with H−1 loads

Abstract Hybrid high-order (HHO) methods for elliptic diffusion problems have been originally formulated for loads in the Lebesgue space $L^2(\varOmega )$. In this paper we devise and analyse a variant thereof, which is defined for any load in the dual Sobolev space $H^{-1}(\varOmega )$. The main feature of the present variant is that its $H^1$-norm error can be bounded only in terms of the $H^1$-norm best error in a space of broken polynomials. We establish this estimate with the help of recent results on the quasi-optimality of nonconforming methods. We prove also an improved error bound in the $L^2$-norm by duality. Compared to previous works on quasi-optimal nonconforming methods the main novelties are that HHO methods handle pairs of unknowns and not a single function and, more crucially, that these methods employ a reconstruction that is one polynomial degree higher than the discrete unknowns. The proposed modification affects only the formulation of the discrete right-hand side. This is obtained by properly mapping discrete test functions into $H^1_0(\varOmega )$.

A hybrid high‐order formulation for a Neumann problem on polytopal meshes

AbstractIn this work, we study a hybrid high‐order (HHO) method for an elliptic diffusion problem with Neumann boundary condition. The proposed method has several features, such as: (a) the support of arbitrary approximation order polynomial at mesh elements and faces on polytopal meshes, (b) the design of a local (element‐wise) potential reconstruction operator and a local stabilization term, that weakly enforces the matching between local element‐ and face‐based on degrees of freedom, and (c) cheap computational cost, thanks to static condensation and compact stencil. We prove the well‐posedness of our HHO formulation, and obtain the optimal error estimates, according to previous study. Implementation aspects are thoroughly discussed. Finally, some numerical examples are provided, which are in agreement with our theoretical results.

A note on the domain mapping method with rough diffusion coefficients

In this article, we consider elliptic diffusion problems on random domains with non-smooth diffusion coefficients. We start by illustrating the problems that arise from a non-smooth diffusion coefficient by recapitulating the corresponding regularity analysis. Then, we propose an alternative approach to address this problem by means of a perturbation method. Based on the assumption that the diffusion coefficient can be decomposed into a possibly deterministic, analytic part and a rough random perturbation, we derive approximation results in terms of the perturbations amplitude for the approximation of quantities of interest of the solution. Numerical examples are given in order to validate and quantify the theoretical results.

Multilevel methods for uncertainty quantification of elliptic PDEs with random anisotropic diffusion

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen–Loeve expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen–Loeve expansion entirely determines the regularity of the solution’s dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution’s mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory.

Isogeometric analysis on non-matching segmentation: discontinuous Galerkin techniques and efficient solvers

One challenging task in Isogeometric Analysis (IGA) is the segmentation of domains, which are represented via their boundaries, e.g. arising from a computer aided design model, into suitable blocks and the construction of their spline parametrizations. This yields a multipatch description of the domain. However, errors can occur during the computation of the associated control points when constructing the spline parametrizations of a patch. This leads to the derivation of non-matching interface parametrizations, i.e. gap and overlapping regions can appear between the patch interfaces. In this paper, we develop discontinuous Galerkin techniques in the context of IGA for solving elliptic diffusion problems on multipatch domains which can include gap and overlapping regions. This work is the continuation of two previous works, where we presented a complete discretization analysis for the case of having only gap regions. Here we focus on a unified presentation of the method for general gap and overlapping regions, and the development of efficient domain decomposition solvers based on tearing and interconnecting technology. In addition, we investigate the influence of the size of the gap/overlapping regions on the performance of the proposed solvers. Comparisons are shown and implementation issues are discussed on actual numerical examples.

Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method

This paper addresses optimization problems constrained by partial differential equations with uncertain coefficients. In particular, the robust control problem and the average control problem are considered for a tracking type cost functional with an additional penalty on the variance of the state. The expressions for the gradient and Hessian corresponding to either problem contain expected value operators. Due to the large number of uncertainties considered in our model, we suggest evaluating these expectations using a multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that this results in the gradient and Hessian corresponding to the MLMC estimator of the original cost functional. Furthermore, we show that the use of certain correlated samples yields a reduction in the total number of samples required. Two optimization methods are investigated: the nonlinear conjugate gradient method and the Newton method. For both, a specific algorithm is provided that dynamically decides which and how many samples should be taken in each iteration. The cost of the optimization up to some specified tolerance $\tau$ is shown to be proportional to the cost of a gradient evaluation with requested root mean square error $\tau$. The algorithms are tested on a model elliptic diffusion problem with lognormal diffusion coefficient. An additional nonlinear term is also considered.

A Duality-Based Optimization Approach for Model Adaptivity in Heterogeneous Multiscale Problems

This paper introduces a novel framework for model adaptivity in the context of heterogeneous multiscale problems. The framework is based on the idea to interpret model adaptivity as a minimization problem of local error indicators that are derived in the general context of the dual weighted residual (DWR) method. Based on the optimization approach a postprocessing strategy is formulated that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. This allows for the systematic, “goal-oriented” tuning of effective models} with respect to a quantity of interest. The framework is tested numerically on elliptic diffusion problems with different types of heterogeneous, random coefficients, as well as an advection-diffusion problem with a strong microscopic, random advection field.

Higher-Order Quasi-Monte Carlo for Bayesian Shape Inversion

In this article, we consider a Bayesian approach towards data assimilation and uncertainty quantification in diffusion problems on random domains. We provide a rigorous analysis of parametric regularity of the posterior distribution given that the data exhibit only limited smoothness. Moreover, we present a dimension truncation analysis for the forward problem, which is formulated in terms of the domain mapping method. Having these novel results at hand, we shall consider as a practical example electrical impedance tomography in the regime of constant conductivities. We are interested in computing moments, in particular, expectation and variance, of the contour of an unknown inclusion, given perturbed surface measurements. By casting the forward problem into the framework of elliptic diffusion problems on random domains, we can directly apply the presented analysis. This straightforwardly yields parametric regularity results for the system response and for the posterior measure, facilitating the application of higher-order quadrature methods for the approximation of moments of quantities of interest. As an example of such a quadrature method, we consider here recently developed higher-order quasi-Monte Carlo methods. To solve the forward problem numerically, we employ a fast boundary integral solver. Numerical examples are provided to illustrate the presented approach and validate the theoretical findings.

Uncertainty Quantification for PDEs with Anisotropic Random Diffusion

In this article, we consider elliptic diffusion problems with an anisotropic random diffusion coefficient. We model the notable direction in terms of a random vector field and derive regularity results for the solution's dependence on the random parameter. It turns out that the decay of the vector field's Karhunen--Loeve expansion entirely determines this regularity. The obtained results allow for sophisticated quadrature methods, such as the quasi--Monte Carlo method or the anisotropic sparse grid quadrature, in order to approximate quantities of interest, like the solution's mean or the variance. Numerical examples in three spatial dimensions are provided to supplement the presented theory.

Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales

AbstractThis paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion coefficients. More precisely, in the context of linear elliptic diffusion problems in randomly packed particulate composites, we develop an approach to strictly bound the error in the expectation and second moment of quantities of interest, without ever solving the fine‐scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine‐scale features. Our computations are purely macroscopic, deterministic and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager–Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss–Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation. Copyright © 2016 John Wiley & Sons, Ltd.

Analysis of the domain mapping method for elliptic diffusion problems on random domains

In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Loeve expansion of the domain perturbation field, we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains.

Discontinuous Galerkin Isogeometric Analysis of Elliptic Diffusion Problems on Segmentations with Gaps

We propose a new discontinuous Galerkin (dG) isogeometric analysis (IgA) technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with nonmatching interfaces. Due to an incorrect segmentation procedure, it may happen that the interfaces of adjacent subdomains do not coincide. In this way, gap regions, which are not present in the original physical domain, are created. In this paper, the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains. We apply a multipatch approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem. The last formulation is further modified by deriving interface conditions without the presence of the solution in the gap. Finally, the solution of this modified problem is approximated b...

Unsteady heat conduction involving phase changes for an irregular bubble/particle entrapped in a solid during freezing – An extension of the heat-balance integral method

Temperature distributions in the molten layer and solid with distinct properties around a bubble or particle entrapped in the solid during unidirectional solidification are determined by applying a heat-balance integral approximation method. The present model can be used to simulate growth, entrapment or departure of a bubble or particle inclusion in solids encountered in manufacturing and materials processing, MEMS, contact melting processes, drilling, etc. In this work, the proposed heat-balance equations are derived by integrating unsteady elliptic heat diffusion equations and introducing the Stefan boundary condition. Due to the time-dependent irregular shapes of phases, coefficients of assumed quadratic temperature profiles are considered to be functions of longitudinal coordinate and time. Temperature coefficients in distinct regions therefore are determined by solving equations governing temperature coefficients derived from heat-balance equations, imposing boundary conditions, and introducing a fictitious boundary condition. The computed temperature fields show agreement with predictions from the finite-difference method. Since the number of independent variables is reduced by one, this work provides an effective method to solve unsteady elliptic diffusion problems experiencing solid–liquid phase changes in irregular shapes.

Vectorization of Diffusion Computations in the Presence of Periodic Boundary Conditions

Effective utilization of translational or rotational periodic boundary conditions, when applicable, can substantially reduce the cost of solving very large multidimensional elliptic diffusion problems. Application of periodic boundary conditions, however, perturbs the overall matrix structure of the underlying discretized diffusion equations, and special care should be exercised to avoid loss of computational efficiency.For simplicity, only the numerical solution of two-dimensional diffusion problems is discussed. Developing and testing on a vector computer alternative algorithms for implementing periodic boundary conditions within the framework of point and line iteration methods are described. For illustration, only the point Chebyshev and red-black line cyclic Chebyshev iterative methods are considered. Vectorization methods previously developed are extended to allow for periodic boundary conditions. The method of odd-even cyclic reduction as applied to vectorization of the solution of tridiagonal systems is generalized to apply to special matrix equations that are almost of tridiagonal form. Consequently, it is demonstrated numerically on a CYBER 205 computer for model two-dimensional problems that the resulting red-black line cyclic Chebyshev iterative method is computationally superior to the highly vectorizable point Chebyshev iterative method. The superiority of the red-black line methods over the point methods is expected to hold for more complex problems with general mesh triangulations.