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Bounds-Constrained Finite Element Approximation of Time-Dependent Partial Differential Equations
Robert C. Kirby, John D. Stephens
TL;DR
Finite element approximations do not generally enforce bounds constraints inherited from the original PDE, while linear spatial and temporal methods face accuracy barriers. The paper proposes projective and monolithic approaches that enforce bounds at discrete time levels, with numerical examples for heat, advection, and Allen–Cahn equations.
Problem
Finite element discretizations can violate PDE bounds constraints, and linear bounds-preserving methods face spatial and temporal order barriers.
Method
The paper combines arbitrary abstract time-stepping schemes with nonlinear projections or reformulates single-stage Runge–Kutta and multistep methods as variational inequalities.
Results
The numerical examples produce bounds-constrained solutions, with projective methods preserving accuracy order and monolithic methods enforcing constraints during the discretized solve.
Takeaways & Limitations
Projective methods provide bounds-constrained solutions while reusing scalable algebraic solvers, whereas monolithic methods address problems with nonlinearities sensitive to bounds constraints.
Takeaways & Limitations
The practical bounds representations are workable but inexact because determining positivity of a multivariate polynomial is NP-hard, and nodal positivity does not guarantee positivity between degrees of freedom.
Abstract
from arXiv · showhide
Finite elements provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not, in general, guarantee enforcement of bounds constraints inherent in the original problem. We propose two approaches to enforcing bounds constraints for time-dependent problems. First, we propose general projective methods which result from a systematic modification of any abstract time-stepping scheme. Second, we present a monolithic technique for which we take a modified formulation of implicit single-stage Runge-Kutta methods and general implicit multistep methods as prototypical examples. By solving a constrained optimization problem, we are able to ensure that the bounds constraints are enforced by the approximate solution at the discrete time levels, obtaining (formally) high order methods both in space and time. Numerical examples for the linear heat and advection equations and nonlinear Allen-Cahn equation are given.
1. Introduction.
The paper addresses bounds violations in time-dependent finite element discretizations by proposing projective and monolithic enforcement approaches. This responds to accuracy barriers and the limited applicability of existing bounds-preserving methods.
- Bounds-preserving discretization must address spatial and temporal approximation simultaneously because linear methods face order barriers.Godunov’s result limits linear monotone single-step schemes for model convection to first order, while positivity-preserving finite element operators are limited to reproducing linear polynomials.
- Projective schemes postprocess a fully discrete finite element and time-stepping approximation with a nonlinear projection onto a bounds-satisfying set.The approach can combine finite elements with Runge–Kutta or multistep methods.
- Monolithic schemes replace the spatial variational equation for the new solution with a variational inequality that enforces the constraints.The formulation captures single-stage Runge–Kutta and multistep methods.
- Existing bounds-enforcement approaches include proximal Galerkin methods, convex optimization-based filtering, and linear programming for Runge–Kutta stage weights.These methods differ in formulation and in how they preserve bounds or accuracy.
- Invariant-domain-preserving discretizations capture deep conservation-law structure but depend strongly on equation structure and may not extend beyond hyperbolic problems.The proposed methods capture less structure but apply to a broader class of PDEs.
2. Problem Setting.
The paper formulates heat, advection, and Allen–Cahn model problems in finite element spaces while constructing discretely feasible sets through basis-dependent bounds on degrees of freedom. Lagrange constraints enforce bounds at nodes, whereas Bernstein constraints enforce uniform bounds but represent only a subset of admissible polynomials.
- Spatial discretization: Finite element discretizations may use conforming or discontinuous spaces, with Fh incorporating mesh-dependent terms such as discontinuous-Galerkin jump terms.The conforming method is recovered when Vh ⊂ V and Fh,t = Ft.
- Model problems: The model problems include heat, divergence-free advection, and nonlinear Allen–Cahn equations with prescribed initial or boundary conditions.The Allen–Cahn solution is constrained by −1 < u < 1 for positive times.
- Feasible sets: Bounds-constrained finite element sets are defined by requiring coefficients in a chosen basis to lie in a prescribed interval or half-open interval.The construction applies to both continuous and discontinuous polynomial spaces.
- Lagrange basis: Lagrange constraints enforce bounds at interpolating nodes, but admissible polynomials can violate those bounds between nodes.Thus the nodally constrained set includes every uniformly admissible polynomial and additional inadmissible ones.
- Bernstein basis: Bernstein constraints produce uniformly bounds-satisfying polynomials through the convex-hull property, but exclude some uniformly admissible polynomials.The paper therefore focuses on nodally constrained Lagrange and uniformly constrained Bernstein bases.
- Approaches: The paper introduces projective methods and monolithic variational-inequality discretizations for enforcing bounds constraints.These approaches modify time-dependent finite element schemes while preserving a basis-defined feasible set.
3. Method Development.
The method development abstracts Runge–Kutta and multistep time steppers into a common update map, then enforces feasibility either by projection or by replacing the update equation with a variational inequality. Projective methods are broadly composable and retain unconstrained solvers, while monolithic methods are more complex but may be more robust for nonlinear problems.
- Time-stepping abstraction: Runge–Kutta and multistep schemes are represented through a common map Φ that computes the next approximation from one or more prior values.The framework uses s = 1 for Runge–Kutta schemes and s > 1 for multistep schemes.
- Projection: A projection onto a closed convex feasible set is equivalently characterized by a constrained variational inequality or constrained minimization problem.The projection is the unique feasible point minimizing distance to the unconstrained value.
- Projective methods: Projective methods compose any supported time discretization with nonlinear projection, producing a discretely feasible approximation at each time level.They are non-invasive, so fast solvers for the unconstrained method remain directly applicable.
- Monolithic methods: Monolithic methods replace the variational equation for the new solution with a variational inequality over the feasible set.The construction covers implicit single-stage Runge–Kutta methods and general multistep methods, including implicit midpoint and BDF(2).
- Comparison: Projective and monolithic methods yield similar numerical solutions for linear problems, while projective methods are significantly faster and monolithic methods may be more robust for nonlinear problems.Monolithic methods require more complex discrete variational inequalities, and unconstrained solvers or preconditioners may not apply directly.
- Additional constraints: Bounds enforcement can be combined with conserved quantities such as mass by enlarging the feasible set, but this complicates implementation and optimization.The additional constraints require a solver that supports them.
4. Some notes on analysis.
The analysis uses projection nonexpansivity to derive error and stability results under increasingly general assumptions. The cleanest error conclusion applies when the semidiscrete system and exact solution are already discretely bounds-feasible.
- Projection analysis: The analysis relies on the nonexpansivity of projections onto closed and convex sets.This property controls the distance between projected approximations and projected exact solutions.
- Error estimates: When the exact solution remains in the discrete feasible set, the constrained method accumulates no more error after each step than the unconstrained method.Established estimates for the unconstrained time stepper therefore apply immediately in this limited case.
- Non-feasible spatial discretization: When the spatial discretization is not bounds-preserving, a separate analysis is required because the exact solution need not belong to the discrete feasible set.The resulting estimate is therefore not covered by the simpler feasible-solution argument.
- Stability: If 0 belongs to the feasible set and the unconstrained method is stable, the projective method is stable in the same sense.A similar stability result holds for monolithic backward Euler, while further stability and error analysis remains future work.
- Numerical scope: Numerical experiments use RadauIIA, Gauss–Legendre, and BDF families with either Lagrange or Bernstein finite elements.The labeling distinguishes the basis family from the polynomial degree.
5. Numerical Results.
The numerical experiments demonstrate that projective and monolithic bounds-constrained finite element methods enforce constraints at discrete times, while convergence and efficiency depend on the spatial basis and time-stepping formulation. The experiments cover heat, advection, and Allen–Cahn problems, with limitations for between-degree-of-freedom bounds and higher-order Lagrange elements.
- Heat equation: Post-processing removes significant undershoots after an unconstrained step, whereas monolithic enforcement obtains the constrained value without computing an intermediate approximation.The comparison uses B6-RIIA(1)-P and B6-RIIA(1)-VI for the heat equation.
- Heat equation: Positivity enforced only at Lagrange degrees of freedom does not guarantee bounds between them, for either post-processing or monolithic methods.The reported violations occur for the initial condition and approximations at t = 0.25.
- Convergence Rates: Optimal-order convergence is observed for the tested bounds-preserving BDF(s) methods, while monolithic RIIA(1)-VI and GL(1)-VI converge at or near the optimal spatial rate in the tested regime.The authors attribute the faster-than-expected Runge–Kutta rates to spatial error dominating temporal error.
- Advection equation: For advection, implicit methods can achieve accuracy and stability with larger time steps, and some fully implicit-plus-projection configurations are faster than unconstrained SSPRK(3, 3).Monolithic constrained systems take significantly longer but avoid intermediate unconstrained approximations.
- Convergence Rates: The constrained methods use Bernstein or Lagrange spatial parameterizations, with Bernstein important for uniformly enforcing bounds and avoiding logarithmic singularities in the nonlinear solver.The experiments report no meaningful timing differences between Bernstein and Lagrange bases for the tested post-processing scaling.
- Spinodal Decomposition: The Allen–Cahn experiment observes energy dissipation in practice for B2-BDF(4)-VI, although the method's energy dissipation is not proved.The simulation uses 20,000 elements, final time T = 10.0, and steps of size k = 1/2.
6. Conclusions and Future Work.
The paper develops projective and monolithic approaches for enforcing bounds constraints in time-dependent finite element methods. Numerical examples demonstrate bounds-constrained computations, while future work targets enforcement between discrete time levels.
- Energy profiles are reported for Bs-BDF(s)-VI with s = 1, 2, 3, plotting values at whole times and every time step.The corresponding snapshots use B2-BDF(2)-VI for the evolution with the stated initial condition.
- Projective methods combine standard time stepping with nonlinear projection onto a discrete feasible set.The approach preserves observed order accuracy and permits reuse of scalable solvers for unconstrained algebraic systems.
- Monolithic methods recast single-stage and multistep variational equations as variational inequalities that enforce constraints during solution.They are more expensive than projective methods but address problems with nonlinearities sensitive to bounds.
- Future Work: The methods enforce bounds constraints at discrete time levels, whereas higher-order methods may also require enforcement between those levels.Future work extends the monolithic approach to implicit collocation-type Runge-Kutta methods through a stage-coupled constrained variational inequality.
- Future Work: Future work seeks uniform temporal-domain enforcement, enabling finer control of bounds constraints and a new class of constrained integration methods.The proposed extension recasts the entire stage-coupled system and uses a novel reformulation.