Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes

We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive definite for a large class of problems, but in general nonsymmetric. We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection–diffusion–reaction problems. Besides the theoretical relationship between the two methods, the results allow for important computational savings in the mixed finite element method, which we finally illustrate on a set of numerical experiments

Four closely related equilibrated flux reconstructions for nonconforming finite elements

International audienceWe 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.Quatre reconstructions très proches de flux équilibrés pour les éléments finis non conformes. Nous étudions la méthode des éléments finis non conformes de Crouzeix et Raviart pour l'équation de Laplace. Nous introduisons quatre reconstructions équilibrées du flux, par prescription directe ou par une approximation mixte de problèmes locaux de Neumann, soit sur le maillage simplectique de départ, soit sur un maillage dual. Nous montrons que toutes ces reconstructions coïncident si le système d'équations linéaires associé est résolu exactement. Nous considérons enfin une solution algébrique inexacte, ajustons les reconstructions du flux et indiquons les différences entre les reconstructions. Pour citer cet article : A. Ern, M. Vohral'ık, C. R. Acad. Sci. Paris, Ser. I 340 (2012)

Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations

International audienceWe present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme

Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs

International audienceWe consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations of diffusion type. To solve these systems, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. We derive adaptive stopping criteria for both iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error significantly. Our estimates also yield a guaranteed upper bound on the overall error at each step of the nonlinear and linear solvers. We prove the (local) efficiency and robustness of the estimates with respect to the size of the nonlinearity owing, in particular, to the error measure involving the dual norm of the residual. Our developments hinge on equilibrated flux reconstructions and yield a general framework. We show how to apply this framework to various discretization schemes like finite elements, nonconforming finite elements, discontinuous Galerkin, finite volumes, and mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers. Numerical experiments for the $p$-Laplacian illustrate the tight overall error control and important computational savings achieved in our approach

hphp-adaptation driven by polynomial-degree-robust a posteriori error estimates for elliptic problems

International audienceWe devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions, for spatially-varying polynomial degree, and for mixed rectangular-triangular grids possibly containing hanging nodes. They deliver a global error upper bound with constant one and, up to data oscillation, error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound. We numerically asses the estimates and several hp-adaptive strategies using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric (odd degrees), and incomplete variants on non-nested unstructured triangular grids for a smooth solution and uniform refinement. Exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement

Financial Management of the Local Action Groups

Import 04/11/2015Práce přehlednou formou ukáže, na jakém principu vznikají místní akční skupiny, seznámí, z čeho MAS vychází, jaké jsou jejích cíle a jaké nástroje jsou používány pro čerpání dotací. V druhé části charakterizuje hospodaření místní akční skupiny Šumperský venkov, v období 2009-2014This work should show with clear form, what is the principal of origin of Local action groups. It introduce, what the LAGs come from, what is theirs goals and which tools are used for frant receives. In second part show the management and economy of Local action group „Šumperský venkov“, in period 2009-2014.153 - Katedra veřejné ekonomikyvelmi dobř