Service-Learning for Introductory and Developmental Math Students

University students in an introductory mathematics course attended a local middle school to assist in tutoring activities as a service-learning project. This project enhanced their learning in their university course. The activity was considered a part of the course, and students were assigned a grade in the same way that they would be for any other course component such as homework or another project. It was found that helping younger school children changes students’ attitudes about mathematics and increases their motivation to learn mathematics, making the service-learning experience meaningful and valuable to all participants. Keywords: Service-learning, Critical reflection, Introductory mathematics, College algebra

Higher-Order SGFEM for One-Dimensional Interface Elliptic Problems with Discontinuous Solutions

We study a class of enriched unfitted finite element or generalized finite element methods (GFEM) to solve a larger class of interface problems, that is, 1D elliptic interface problems with discontinuous solutions, including those having implicit or Robin-type interface jump conditions. The major challenge of GFEM development is to construct enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. The linear stable generalized finite element method (SGFEM) was recently developed using one enrichment function. We generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. So is the optimal order convergence at all nodes. To prove the efficiency of the SGFEM, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present

Finite Elements and Practical Error Analysis of Huxley and EFK Equations

In this dissertation, long time error estimates are obtained using non-traditional methods for the Hodgkin-Huxley equation ut + uxx = u(1-u)(u-a) for a ∈ (0,1/2) ut + γ u xxxx - uxx = u-u3 Traditional methods for analyzing exact error propagation depends on the stability of the numerical method employed. Whereas, in this dissertation the analysis of the exact error propagation uses evolving attractors and only depends on the stability of the dynamical system. The use of the smoothing indicator yields a posteriori estimates on the numerical error instead of a priori estimates

Finite Elements and Practical Error Analysis of Huxley and EFK Equations

In this dissertation, long time error estimates are obtained using non-traditional methods for the Hodgkin-Huxley equation ut + uxx = u(1-u)(u-a) for a ∈ (0,1/2) ut + γ u xxxx - uxx = u-u3 Traditional methods for analyzing exact error propagation depends on the stability of the numerical method employed. Whereas, in this dissertation the analysis of the exact error propagation uses evolving attractors and only depends on the stability of the dynamical system. The use of the smoothing indicator yields a posteriori estimates on the numerical error instead of a priori estimates

Flux Recovery and Superconvergence of Quadratic Immersed Interface Finite Elements

We introduce a flux recovery scheme for the computed solution of a quadratic immersed finite element method introduced by Lin et al. in [13]. The recovery is done at nodes and interface point first and by interpolation at the remaining points. In the case of piecewise constant diffusion coefficient, we show that the end nodes are superconvergence points for both the primary variable p and its flux u. Furthermore, in the case of piecewise constant diffusion coefficient without the absorption term the errors at end nodes and interface point in the approximation of u and p are zero. In the general case, flux error at end nodes and interface point is third order. Numerical results are provided to confirm the theory

Numerical methods for phase transition problems

Consiglio Nazionale delle Ricerche - Biblioteca Centrale CNR 7, / CNR - Consiglio Nazionale delle RichercheSIGLEITItal