Variable-coefficient partial differential equations (PDEs) are important in making predictions of complex physical, mechanical, and engineering systems. It is still difficult to accurately and efficiently solve such PDEs because their coefficients may have heterogeneities, non linearities, and complicated geometries of the domains. This paper introduces a state-of-the-art finite element model to solve variable coefficient PDEs that use higher-order spatial discretization, high-quality time integration and adaptive mesh techniques. It was demonstrated that the proposed method was accurate on benchmark problems, such as linear and nonlinear diffusion-reaction equations, fractional diffusion-wave equations, and transient NavierStokes-transport systems. Optimal convergence rates are shown by numerical results, i.e. second and third order accuracy with linear and quadratic elements respectively, and local super convergence effects by up to 25 percent decrease in error. The framework was stable to high coefficient variability and the number of solver iterations rose by no more than 60 percent even when coefficient variations were up to 150 percent of the mean coefficient value. The proposed approach has been proven to be more accurate, robust and computationally efficient than the classical finite difference and standard finite element methods. The paper will add to the available body of literature by presenting a versatile approach to finite element modeling, which can be used to solve modeling problems that involve variable-coefficient PDEs in multi-dimensional and time-dependent and nonlinear equations with high confidence.