hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

Computer-based summative assessment of clinical reasoning in clerkships: A mixed-method comparison of key feature problems with case-based multiple choice questions

Background: It is yet unclear if there are differences between using electronic key feature problems (KFPs) or electronic case-based multiple choice questions (cbMCQ) for the assessment of clinical decision making. Summary of Work: Fifth year medical students were exposed to clerkships which ended with a summative exam. Assessment of knowledge per exam was done by 6-9 KFPs, 9-20 cbMCQ and 9-28 MC questions. Each KFP consisted of a case vignette and three key features (KF) using “long menu” as question format. We sought students’ perceptions of the KFPs and cbMCQs in focus groups (n of students=39). Furthermore statistical data of 11 exams (n of students=377) concerning the KFPs and (cb)MCQs were compared. Summary of Results: The analysis of the focus groups resulted in four themes reflecting students’ perceptions of KFPs and their comparison with (cb)MCQ: KFPs were perceived as (i) more realistic, (ii) more difficult, (iii) more motivating for the intense study of clinical reasoning than (cb)MCQ and (iv) showed an overall good acceptance when some preconditions are taken into account. The statistical analysis revealed that there was no difference in difficulty; however KFP showed a higher discrimination and reliability (G-coefficient) even when corrected for testing times. Correlation of the different exam parts was intermediate. Conclusions: Students perceived the KFPs as more motivating for the study of clinical reasoning. Statistically KFPs showed a higher discrimination and higher reliability than cbMCQs. Take-home messages: Including KFPs with long menu questions into summative clerkship exams seems to offer positive educational effects.