Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed metho

Functional-type a posteriori error estimates for mixed finite element methods

This paper concerns a posteriori error estimation for the primal and dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting with inhomogeneous mixed Dirichlet–Neumann boundary conditions. New functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to ensure upper bounds for discretization errors in primal and dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart–Thomas or continuous piecewise linear elements. It is shown how these projections can be easily realized by simple local averaging

A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions

The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive the reliable and efficient a posteriori error estimator that measures the error in the energy norm. The estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H1, independently of the discretization method chosen. In particular, our error estimator can be applied also to problems and discretizations where the Galerkin orthogonality is not available. We will present different strategies for the evaluation of the error estimator. Only one constant appears in its definition which is the one from Friedrichs' inequality; that constant depends solely on the domain geometry, and the estimator is quite non-sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation

Particle transport method for convection problems with reaction and diffusion

The paper is devoted to the further development of the particle transport method for the convection problems with diffusion and reaction. Here, the particle transport method for a convection-reaction problem is combined with an Eulerian finite-element method for diffusion in the framework of the operator-splitting approach. The technique possesses a special spatial adaptivity to resolve solution singularities possible due to convection and reaction terms. A monotone projection technique is used to transfer the solution of the convection-reaction subproblem from a moving set of particles onto a fixed grid to initialize the diffusion subproblem. The proposed approach exhibits good mass conservation and works with structured and unstructured meshes. The performance of the presented algorithm is tested on one- and two-dimensional benchmark problems. The numerical results confirm that the method demonstrates good accuracy for the convection-dominated as well as for convection-diffusion problems. Copyright © 2007 John Wiley & Sons, Ltd

High Throughput, High Resolution Enzymatic Lithography Process: Effect of Crystallite Size, Moisture, and Enzyme Concentration

By bringing enzymes into contact with predefined regions of a surface, a polymer film can be selectively degraded to form desired patterns that find a variety of applications in biotechnology and electronics. This so-called “enzymatic lithography” is an environmentally friendly process as it does not require actinic radiation or synthetic chemicals to develop the patterns. A significant challenge to using enzymatic lithography has been the need to restrict the mobility of the enzyme in order to maintain control of feature sizes. Previous approaches have resulted in low throughput and were limited to polymer films only a few nanometers thick. In this paper, we demonstrate an enzymatic lithography system based on Candida antartica lipase B (CALB) and poly­(ε-caprolactone) (PCL) that can resolve fine-scale features, (<1 μm across) in thick (0.1–2.0 μm) polymer films. A Polymer Pen Lithography (PPL) tool was developed to deposit an aqueous solution of CALB onto a spin-cast PCL film. Immobilization of the enzyme on the polymer surface was monitored using fluorescence microscopy by labeling CALB with FITC. The crystallite size in the PCL films was systematically varied; small crystallites resulted in significantly faster etch rates (20 nm/min) and the ability to resolve smaller features (as fine as 1 μm). The effect of printing conditions and relative humidity during incubation is also presented. Patterns formed in the PCL film were transferred to an underlying copper foil demonstrating a “Green” approach to the fabrication of printed circuit boards