Nonlinearly-Stable High-Order Methods on Simplices with Improved Efficiency

The expression ( Q + Q^T ) represents a matrix where ( Q ) is nearly skew-symmetric. By applying a similar approach as in the continuous case, we find that ( H ) serves as an approximation or quadrature. When we multiply this by the solution and integrate over the domain discretely, we replicate the process used in the continuous case. Utilizing the properties of ( Q ), we derive a form that corresponds to the discrete version of the equation.

Additionally, the total approximated integral of ( u^2 ) within the domain is determined solely by the values at the boundaries, specifically by what enters and exits the domain.