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Finite element simulations have been used to solve various partialdifferential equations (PDEs) that model physical, chemical, and biologicalphenomena. The resulting discretized solutions to PDEs often do not satisfyrequisite physical properties, such as positivity or monotonicity. Such invalidsolutions pose both modeling challenges, since the physical interpretation ofsimulation results is not possible, and computational challenges, since suchproperties may be required to advance the scheme. We, therefore, consider theproblem of computing solutions that preserve these structural solutionproperties, which we enforce as additional constraints on the solution. Weconsider in particular the class of convex constraints, which includespositivity and monotonicity. By embedding such constraints as a postprocessingconvex optimization procedure, we can compute solutions that satisfy generaltypes of convex constraints. For certain types of constraints (includingpositivity and monotonicity), the optimization is a filter, i.e., anorm-decreasing operation. We provide a variety of tests on one-dimensionaltime-dependent PDEs that demonstrate the method's efficacy, and we empiricallyshow that rates of convergence are unaffected by the inclusion of theconstraints.

Structure-preserving Nonlinear Filtering for Continuous and Discontinuous Galerkin Spectral/hp Element Methods