This paper studies the efficiency of numerical homogenization methods, specifically the finite element heterogeneous multiscale method (FE-HMM), for solving second-order elliptic problems with highly oscillatory coefficients. The problem is of the form $ -\nabla\cdot\left(a^{\varepsilon}(x)\nabla u^{\varepsilon}(x)\right)=f(x)\mathrm{in}\Omega,\qquad u^{\varepsilon}(x)=0\mathrm{on}\partial\Omega $, where $ a^{\varepsilon}(x) $ is a tensor that varies on a small scale $ \varepsilon $. The FE-HMM computes an effective solution $ u^{0}(x) $ by using a macro FE space and micro FEMs on sampling domains. The method is efficient as it reduces computational cost by using proper averaging of micro solutions and numerical correctors. A fully discrete a priori convergence analysis shows that the complexity of the FE-HMM is superlinear with respect to the macroscopic degrees of freedom. However, for high-dimensional problems or high-order macro methods, the FE-HMM can become costly. Three ways to reduce the complexity are discussed: (1) using pseudo-spectral methods for regular micro oscillations, (2) using reduced basis framework for regular dependence on macro variables, and (3) using adaptive mesh refinement for low regularity of the effective solution. The paper also acknowledges the support from the Swiss National Science Foundation.This paper studies the efficiency of numerical homogenization methods, specifically the finite element heterogeneous multiscale method (FE-HMM), for solving second-order elliptic problems with highly oscillatory coefficients. The problem is of the form $ -\nabla\cdot\left(a^{\varepsilon}(x)\nabla u^{\varepsilon}(x)\right)=f(x)\mathrm{in}\Omega,\qquad u^{\varepsilon}(x)=0\mathrm{on}\partial\Omega $, where $ a^{\varepsilon}(x) $ is a tensor that varies on a small scale $ \varepsilon $. The FE-HMM computes an effective solution $ u^{0}(x) $ by using a macro FE space and micro FEMs on sampling domains. The method is efficient as it reduces computational cost by using proper averaging of micro solutions and numerical correctors. A fully discrete a priori convergence analysis shows that the complexity of the FE-HMM is superlinear with respect to the macroscopic degrees of freedom. However, for high-dimensional problems or high-order macro methods, the FE-HMM can become costly. Three ways to reduce the complexity are discussed: (1) using pseudo-spectral methods for regular micro oscillations, (2) using reduced basis framework for regular dependence on macro variables, and (3) using adaptive mesh refinement for low regularity of the effective solution. The paper also acknowledges the support from the Swiss National Science Foundation.