This mixed-methods study explores the interconnected influence of growth mindset, mindfulness, and second language (L2) self-efficacy on language achievement in intermediate English as a foreign language (EFL) learners. Quantitative data (n = 411) analyzed through Structural Equation Modeling (SEM) and qualitative insights from semi-structured interviews (n = 19) reveal significant positive associations between all three factors and L2 achievement. SEM further clarifies the direct and indirect pathways through which these psychological elements impact learning outcomes. Qualitative findings enrich this understanding by showcasing learners' lived experiences and the transformative power of these variables in fostering a supportive learning environment. Collectively, the results emphasize the importance of integrating growth-oriented beliefs, nurturing self-efficacy, and employing mindfulness practices in language education to optimize learning. This research offers notable insights for second language acquisition and educational psychology, informing educators, policymakers, and practitioners about effective strategies for promoting successful language learning journeys.

Standard discontinuous Galerkin methods, based on piecewise polynomials of degree q = 0, 1, are considered for temporal semi-discretization for second-order hyperbolic equations. The main goal of this paper is to present a simple and straightforward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved. To this end, energy identities and stability estimates of the discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical results.

Discontinuous Galerkin methods, based on piecewise polynomials of degree $q\geq 0$, are investigated for temporal semi-discretization for second order hyperbolic equations. Energy identities and stability estimates of the discrete problem are proved for a slightly more general problem, that are used to prove optimal order a priori error estimates with minimal regularity requirement. Uniform norm in time error estimates are proved for the constant and linear cases. Numerical experiments are performed to verify the theoretical rate of convergence.