This work presents a priori and a posteriori error analyses of a new multiscale hybridmixed method (MHM) for an elliptic model. Specially designed to incorporate multiple scales into the construction of basis functions, this finite element method relaxes the continuity of the primal variable through the action of Lagrange multipliers, while assuring the strong continuity of the normal component of the flux (dual variable). As a result, the dual variable, which stems from a simple postprocessing of the primal variable, preserves local conservation. We prove existence and uniqueness of a solution for the MHM method as well as optimal convergence estimates of any order in the natural norms. Also, we propose a face-residual a posteriori error estimator, and prove that it controls the error of both variables in the natural norms. Several numerical tests assess the theoretical results.

The aim of this paper is to introduce residual type a posteriori error estimators for a Poisson problem with a Dirac delta source term, in L p norm and W 1,p seminorm. The estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.

This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary. On the other hand, since the test function space is enriched with bubble-like functions, a Petrov-Galerkin approach is employed. We use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear/piecewise constant interpolation pairs. Optimal order convergence results are derived and numerical tests validate the proposed methods. Introduction.Finite element solution of the Stokes problem poses the basic problem of satisfying the discrete Babuska-Brezzi (or inf-sup) condition (see [24] and the references therein). This is indeed a restriction from the point of view of implementation since equal order velocity and pressure spaces do not satisfy this condition. On the other hand, the minimal space to imagine, namely continuous piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure, does not satisfy this condition either.Several solutions have been proposed to overcome this restriction, starting with that in [11] and the first consistent method in [28]. Moreover,in [23,27,29,34] the possibility of considering discontinuous spaces for the pressure was considered and justified. On the other hand, in [14,13], the idea from [16] has been used to propose a new kind of stabilized finite element methods, with stabilizing terms now containing only jump terms across the interelement boundaries. For an overview of stabilized finite element methods for the Stokes problem, see [19] and [5].On the other hand, the theoretical justification of stabilized methods has become a subject of interest in the last decade. In [2,3,4,31], the connection between stabilized finite element methods and Galerkin methods enriched with bubble functions has been used to propose new stabilized finite element methods for Stokes-like and linearized Navier-Stokes problems. Also, in [22] macro bubbles were used to derive a method analogous to the locally stabilized method from [29] containing jump terms across the interelement boundaries of the macroelements. In the resulting method, the stabilizing

An adaptive finite element scheme for the advection-reaction-diffusion equation is introduced and analyzed. This scheme is based on a stabilized finite element method combined with a residual error estimator. The estimator is proved to be reliable and efficient. More precisely, global upper and local lower error estimates with constants depending at most on the local mesh Peclet number are proved. The effectiveness of this approach is illustrated by several numerical experiments.

Slab dip controls the state of stress in an overriding plate and affects mountain building. Analog and numerical models have shown variations in tectonic regime induced by slab folding over the 660 km depth discontinuity zone in orthogonal convergence. Here using a three‐dimensional model of oblique subduction (30°) and accounting for free top surfaces, we show how slab folding generates an along‐strike slab dip segmentation, inducing variations in topography of the overriding plate. When the subducting plate begins to curve forward, the elevation height rises inland and varies along the trench from 5 km to 2 km. The Andes are a suitable natural zone to compare our results with because of its linear margin and well‐constrained plates kinematics. Thus, we provide a new explanation to the general decrease in elevation from the central to southern Andes, which still remains to be combined with other 3‐D mechanisms to explain the actual Andean topography.

The multiscale hybrid-mixed (MHM) method is extended to the Stokes and Brinkman equations with highly heterogeneous coefficients. The approach is constructive. We first propose an equivalent dual-hybrid formulation of the original problem using a coarse partition of the heterogeneous domain. Faces may be not aligned with jumps in the data. Then, the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent Stokes (or Brinkman) problems at the continuous level. Owing to this decomposition, the one-level MHM method stems from the standard Galerkin approach for the Lagrange multiplier space. Basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation. They are the exact solution of the local problems with prescribed Neumann boundary conditions on faces driven by the Lagrange multipliers. We make the MHM method effective by adopting the unusual stabilized finite element method to solve the local problems approximately. As such, equal-order interpolation turns out to be an option for the velocity, the pressure and the Lagrange multipliers. The numerical solutions share the important properties of the continuum, such as local equilibrium with respect to external forces and the local mass conservation. Several academic and highly heterogeneous tests infer that the method achieves super-convergence for the velocity as well as optimal convergence for the pressure and also for the stress tensor in their natural norms.

In this work we introduce a new a posteriori error estimate of hierarchical type for the advection-diffusion-reaction equation. We prove the equivalence between the energy norm of the error and our error estimate using an auxiliary linear problem for the residual and an easy way to prove inf–sup condition.

Residual type a posteriori error estimators are introduced in this paper for an advection-diffusion-reaction problem with a Dirac delta source term. The error is measured in an adequately weighted W 1,p -norm. These estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.