Anisotropic mesh refinement in polyhedral domains: error estimates with data in L^2(\Omega)

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H^1(\Omega)- and L^2(\Omega)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L^2(\Omega)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equation and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.Comment: 28 pages, 7 figure

Analysis of finite element methods for singularly perturbed problems

Proponemos y analizamos métodos de elementos finitos para problemas estacionarios singularmente perturbados, tales como problemas de reacción-difusión o de convección-difusión. Es conocido que la técnicas de discretización estándares no producen buenas aproximaciones a la solución de esta clase de problemas si el parámetro de perturbación es pequeño debido a la presencia de capas límites o internas. Estamos interesados en métodos robustos que funcionen adecuadamentepara todos los valores del parámetro de perturbación singular. Consideramos dos técnicas diferentes. Una de ellas se basa en refinamientos locales de la malla cerca de las capas límites. Usamos que la solución está en espacios de Sobolev con peso para probar estimaciones del error de interpolación sobre mallas rectangulares adecuadamente graduadas. Introducimos un operadorde interpolación de promedios para el cual probamos estimaciones de error bajo la condición de que elementos vecinos tengan longitudes comparables en cadadirección. Esta condición es verificada por mallas que aparecen naturalmente en la aproximación de capas límites. También consideramos la aproximación defunciones que se anulan en el borde por funciones con la misma propiedad. Finalmente, nuestras estimaciones permiten sobre el lado derecho normas de Sobolev con pesos, donde el peso está relacionado con la distancia al borde. Proponemos también un método de Galerkin Discontinuo (DG) con estabilización de tipo Exponential Fitting para resolver un problema de interés en semiconductores. El método DG considerado es una variante del método de Interior Penalty. Analizamos el método propuesto en las formulaciones mixta y primal, y presentamos ejemplos númericos que muestran resultados adecuados. Finalmente probamos estimaciones de error óptimas para el método DG introducido en el caso de un problema regular.We develop and analyze finite element methods for stationary singularly perturbed problems such us reaction-diffusion or convection-diffusion problems. It is known that standard discretization techniques do not give good approximations to the solution of this kind of problems when the perturbation parameter is small because of the presence of boundary or internal layers. We are interested in obtaining robust methods that work for all the values of the singular pertubation parameter. We consider two different finite element techniques. One of them is based on mesh refinements near the boundary layers. We use the fact that the solution is in weighted Sobolev spaces in order to prove interpolacion error estimates on suitably graded rectangular meshes. We prove our error estimates for a mean interpolation operator under the mild condition that neighboring elements have comparable sizes in each direction. This condition is verified for the meshesthat appear naturally in the approximation of boundary layers. Also we consider the aprproximation of function vanishing on the boundary by functions with the same property. Finally, our estimates allow on the right hand side some weighted Sobolev norms where the weight is related with the distance to the boundary. We also propose a Discontinuous Galerkin (DG) method with stabilization of Exponential Fitting type to approximate the solution of a problem of interest in semiconductors. The DG method considered here is a modification of the Interior Penalty method. We analyze the proposed method in mixed and primal formulation paying attention to the presence of "overflow", and we present some numerical examples showing adequate results. Finally we prove optimal error estimates for the DG method introduced here for a regular problem.Fil: Lombardi, Ariel L.. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Interior penalty discontinuous Galerkin FEM for the p(x)p(x)-Laplacian

In this paper we construct an "Interior Penalty" Discontinuous Galerkin method to approximate the minimizer of a variational problem related to the $p(x)-$Laplacian. The function $p:\Omega\to [p_1,p_2]$ is log H\"{o}lder continuous and $1<p_1\leq p_2<\infty$. We prove that the minimizers of the discrete functional converge to the solution. We also make some numerical experiments in dimension one to compare this method with the Conforming Galerkin Method, in the case where $p_1$ is close to one. This example is motivated by its applications to image processing.Comment: 26 pages, 2 figure

Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Abstract Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

Analysis of finite element methods for singularly perturbed problems

Proponemos y analizamos métodos de elementos finitos para problemas estacionarios singularmente perturbados, tales como problemas de reacción-difusión o de convección-difusión. Es conocido que la técnicas de discretización estándares no producen buenas aproximaciones a la solución de esta clase de problemas si el parámetro de perturbación es pequeño debido a la presencia de capas límites o internas. Estamos interesados en métodos robustos que funcionen adecuadamentepara todos los valores del parámetro de perturbación singular. Consideramos dos técnicas diferentes. Una de ellas se basa en refinamientos locales de la malla cerca de las capas límites. Usamos que la solución está en espacios de Sobolev con peso para probar estimaciones del error de interpolación sobre mallas rectangulares adecuadamente graduadas. Introducimos un operadorde interpolación de promedios para el cual probamos estimaciones de error bajo la condición de que elementos vecinos tengan longitudes comparables en cadadirección. Esta condición es verificada por mallas que aparecen naturalmente en la aproximación de capas límites. También consideramos la aproximación defunciones que se anulan en el borde por funciones con la misma propiedad. Finalmente, nuestras estimaciones permiten sobre el lado derecho normas de Sobolev con pesos, donde el peso está relacionado con la distancia al borde. Proponemos también un método de Galerkin Discontinuo (DG) con estabilización de tipo Exponential Fitting para resolver un problema de interés en semiconductores. El método DG considerado es una variante del método de Interior Penalty. Analizamos el método propuesto en las formulaciones mixta y primal, y presentamos ejemplos númericos que muestran resultados adecuados. Finalmente probamos estimaciones de error óptimas para el método DG introducido en el caso de un problema regular.We develop and analyze finite element methods for stationary singularly perturbed problems such us reaction-diffusion or convection-diffusion problems. It is known that standard discretization techniques do not give good approximations to the solution of this kind of problems when the perturbation parameter is small because of the presence of boundary or internal layers. We are interested in obtaining robust methods that work for all the values of the singular pertubation parameter. We consider two different finite element techniques. One of them is based on mesh refinements near the boundary layers. We use the fact that the solution is in weighted Sobolev spaces in order to prove interpolacion error estimates on suitably graded rectangular meshes. We prove our error estimates for a mean interpolation operator under the mild condition that neighboring elements have comparable sizes in each direction. This condition is verified for the meshesthat appear naturally in the approximation of boundary layers. Also we consider the aprproximation of function vanishing on the boundary by functions with the same property. Finally, our estimates allow on the right hand side some weighted Sobolev norms where the weight is related with the distance to the boundary. We also propose a Discontinuous Galerkin (DG) method with stabilization of Exponential Fitting type to approximate the solution of a problem of interest in semiconductors. The DG method considered here is a modification of the Interior Penalty method. We analyze the proposed method in mixed and primal formulation paying attention to the presence of "overflow", and we present some numerical examples showing adequate results. Finally we prove optimal error estimates for the DG method introduced here for a regular problem.Fil: Lombardi, Ariel L.. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

interpolation

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