  In simulation sciences, it is desirable to capture the real-world problem
features as accurately as possible. Methods popular for scientific simulations
such as the finite element method (FEM) and finite volume method (FVM) use
piecewise polynomials to approximate various characteristics of a problem, such
as the concentration profile and the temperature distribution across the
domain. Polynomials are prone to creating artifacts such as Gibbs oscillations
while capturing a complex profile. An efficient and accurate approach must be
applied to deal with such inconsistencies in order to obtain accurate
simulations. This often entails dealing with negative values for the
concentration of chemicals, exceeding a percentage value over 100, and other
such problems. We consider these inconsistencies in the context of partial
differential equations (PDEs). We propose an innovative filter based on convex
optimization to deal with the inconsistencies observed in polynomial-based
simulations. In two or three spatial dimensions, additional complexities are
involved in solving the problems related to structure preservation. We present
the construction and application of a structure-preserving filter with a focus
on multidimensional PDEs. Methods used such as the Barycentric interpolation
for polynomial evaluation at arbitrary points in the domain and an optimized
root-finder to identify points of interest improve the filter efficiency,
usability, and robustness. Lastly, we present numerical experiments in 2D and
3D using discontinuous Galerkin formulation and demonstrate the filter's
efficacy to preserve the desired structure. As a real-world application,
implementation of the mathematical biology model involving platelet aggregation
and blood coagulation has been reviewed and the issues around FEM
implementation of the model are resolved by applying the proposed
structure-preserving filter.
