  Finite element simulations have been used to solve various partial
differential equations (PDEs) that model physical, chemical, and biological
phenomena. The resulting discretized solutions to PDEs often do not satisfy
requisite physical properties, such as positivity or monotonicity. Such invalid
solutions pose both modeling challenges, since the physical interpretation of
simulation results is not possible, and computational challenges, since such
properties may be required to advance the scheme. We, therefore, consider the
problem of computing solutions that preserve these structural solution
properties, which we enforce as additional constraints on the solution. We
consider in particular the class of convex constraints, which includes
positivity and monotonicity. By embedding such constraints as a postprocessing
convex optimization procedure, we can compute solutions that satisfy general
types of convex constraints. For certain types of constraints (including
positivity and monotonicity), the optimization is a filter, i.e., a
norm-decreasing operation. We provide a variety of tests on one-dimensional
time-dependent PDEs that demonstrate the method's efficacy, and we empirically
show that rates of convergence are unaffected by the inclusion of the
constraints.

Numerical Analysis;Numerical Analysis