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A numerical method for a two-dimensional curl-curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive ) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

We study a class of symmetric discontinuous Galerkin methods on graded meshes. Optimal order error estimates are derived in both the energy norm and the L 2 norm, and we establish the uniform convergence of V -cycle, F-cycle and W -cycle multigrid algorithms for the resulting discrete problems. Numerical results that confirm the theoretical results are also presented.

BackgroundFew studies have investigated the prevalence of refractive error (RE) in older adults in China, and most have focused on East China. Our study determined the prevalence and risk factors of RE in Han and Yi adults aged 40–80 years in rural and urban areas in Yunnan Province, Southwest China.MethodsOur cross-sectional study is part of the China National Health Survey (CNHS). The age-adjusted prevalence rates of RE in Han and Yi adults aged 40–80 years in Yunnan were compared. We used a multivariate logistic regression model to identify risk factors for myopia and hyperopia.ResultsAmong 1626 participants, the age-adjusted prevalence rates of myopia, hyperopia, high myopia and astigmatism were 26.35% (95%CI 24.01–28.70%), 19.89% (95%CI 18.16–21.61%), 2.64% (95%CI 1.75–3.53%), and 56.82% (95%CI 54.31–59.34%). Compared to the Yi population, the Han population had higher prevalence of myopia (31.50% vs 16.80%, p < 0.0001), high myopia (3.34% vs 1.31%, p = 0.049) and astigmatism (60.07% vs 50.67%, p = 0.026) but lower prevalence of hyperopia (16.58% vs 27.37%, p < 0.0001). In the multivariate logistic regression, individuals aged 45–49 (p < 0.001), 50–54 (p < 0.001), 55–59 (p = 0.014), and 60–64 years (p = 0.005) had a lower myopia risk than those aged 40–44 years, and individuals aged 50–54 (p = 0.002), 55–59, 60–64 and 65 years and older (all p < 0.001) had a higher hyperopia risk than those aged 40–44 years. Myopia was also associated with height (p = 0.035), time spent in rural areas (p = 0.014), undergraduate/graduate education level (p = 0.001, compared with primary school or lower education level) and diabetes (p = 0.008). The Yi population had a higher risk of hyperopia than the Han population (p = 0.025). Moreover, hyperopia was related to time spent in rural areas (p < 0.001) and pterygium (p = 0.019).ConclusionsOur study investigated the overall prevalence of RE in older adults in rural and urban areas of Southwest China. Compared to the Yi population, the Han population had a higher prevalence of myopia, high myopia and astigmatism but a lower risk of hyperopia. The prevalence of myopia in the Han population in underdeveloped Southwest China was similar to that of residents in East China or of Chinese Singaporeans under urban or rural settings.

AimsTo investigate the prevalence, ethnic differences and associated risk factors for pterygium in Han and Manchu populations aged 40–79 years in Hebei province, China.DesignCross-sectional study, as a part of the China National Health Survey.SettingHebei province, China.ParticipantsA multistage cluster sampling method with urbanisation level-based stratification was used to select participants for this study. A total of 4591 individuals over 40 years were recruited for this study. Inclusive criteria: (1) residents who had been living in Hebei for more than 1 year; (2) Han individuals with both parents being Han, or Manchu individuals with both parents being Manchu; (3) underwent ophthalmic examinations and (4) information in the questionnaire was complete.Main outcome measuresMultiple logistic regression analysis was used to evaluate the association between pterygium prevalence and factors of interest.ResultsA total of 3790 individuals (2351 Hans and 1439 Manchus) met the study criteria, of which 248 were diagnosed with pterygium (6.5%). There was no significant difference between the prevalence rates in Hans (6.2%) and Manchus (7.2%) (p=0.232). Multivariate analysis revealed that the risk factors for grade 2 or higher pterygium were increasing age (p<0.001) and rural residence (OR 1.83; 95% CI 1.11 to 3.02; p=0.018), while the protective factors include gender (female) (OR 0.58; 95% CI 0.37 to 0.88; p=0.011), cigarette smoking (OR 0.53; 95% CI 0.34 to 0.83; p=0.005) and myopia (OR 0.50; 95% CI 0.33 to 0.77; p=0.002). Premature menopause (OR 2.66; 95% CI 1.05 to 6.72; p=0.038) increased the risk of grade 2 or higher pterygium in females, while higher high-density lipoprotein (HDL) (OR 1.94; 95% CI 1.08 to 3.47; p=0.027) was a risk factor of grade 2 or higher pterygium in males.ConclusionThe overall prevalence of pterygium in Han and Manchu population in Hebei, China was approximately 6.1%. There were no differences in the prevalence of pterygium between Hans and Manchus, and the race was not a risk factor. This is the first study to report on the positive association between premature menopause and pterygium in females and between higher HDL levels and pterygium in males.

The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi-optimal error estimates in both the energy norm and the L 2 norm for the SIP method, and prove uniform convergence of the W -cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results. MULTIGRID METHODS FOR THE SYMMETRIC INTERIOR PENALTY METHOD483 Hence, the model problem (1) has a unique solution u for any f ∈ L 2, ( ). Moreover u has the following properties.(i) The second-order weak derivatives of u belong to L 2, and they satisfy

Abstract. We propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Hodge decomposition for divergencefree vector fields. In this approach an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P 1 finite element method.