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High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

Robert C. Kirby, Daniel Shapero

TL;DR

The paper addresses preserving hard bounds (e.g., maximum principles) in finite element discretizations by replacing linear variational problems with discrete variational inequalities. It establishes a Céa-type best-approximation bound for the constrained setting and proves that bounds-constrained approximants can match the accuracy of unconstrained best approximants in the $W^{1,p}$ sense under suitable conditions. A practical pathway is presented via enforcing bounds on Bernstein coefficients, enabling higher-order, bound-respecting polynomials without requiring representation of the entire constrained space. Numerical experiments on diffusion and convection-diffusion problems demonstrate that the variational-inequality approach achieves nonnegativity and sharp feature resolution with high-order discretizations, highlighting potential for robust, bounds-preserving simulations in complex PDEs.

Abstract

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the $W^{1,p}$ norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints.

High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

TL;DR

The paper addresses preserving hard bounds (e.g., maximum principles) in finite element discretizations by replacing linear variational problems with discrete variational inequalities. It establishes a Céa-type best-approximation bound for the constrained setting and proves that bounds-constrained approximants can match the accuracy of unconstrained best approximants in the sense under suitable conditions. A practical pathway is presented via enforcing bounds on Bernstein coefficients, enabling higher-order, bound-respecting polynomials without requiring representation of the entire constrained space. Numerical experiments on diffusion and convection-diffusion problems demonstrate that the variational-inequality approach achieves nonnegativity and sharp feature resolution with high-order discretizations, highlighting potential for robust, bounds-preserving simulations in complex PDEs.

Abstract

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints.
Paper Structure (13 sections, 4 theorems, 60 equations, 8 figures)

This paper contains 13 sections, 4 theorems, 60 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Abstract setting
  4. Examples
  5. Best approximation result
  6. Constrained approximation
  7. Approximation by bounds-constrained polynomials
  8. Representing bounds-constrained polynomials
  9. Numerical results
  10. Diffusion
  11. Stationary convection-diffusion
  12. Time-dependent convection-diffusion
  13. Conclusions

Key Result

Theorem 1

Let $V$ be a Hilbert space with closed, convex subset $U$. Let bilinear form $a$ satisfy eq:cont and eq:coercive and $F \in V^\prime$. Let $u \in V$ satisfy the variational problem eq:vareq as well as $u \in U$. For any finite-dimensional subspace $V_h \subseteq V$ with closed convex subset $U_h \su

Figures (8)

  • Figure 1: Cubic Bernstein polynomials.
  • Figure 2: Error approximating \ref{['eq:ellipticPDE']} versus $N$ on an $N \times N$ mesh for various polynomial degrees. We compare the finite element solution obtained by solving a linear variational problem to the solution of a discrete variational inequality enforcing bounds on the Bernstein coefficients. In each case, we observe similar convergence rates for the solution by each technique.
  • Figure 3: Plots of solution to the variational problem, variational inequality, and their difference using quadratic Bernstein polynomials on a $16 \times 16$ mesh. Notably, the variational inequality gives a uniformly nonnegative solution, unlike the solution of the variational problem.
  • Figure 4: Solutions of the variational problem (VP), variational inequality (VI), and their difference using linear, quadratic, and cubic basis functions with forcing function \ref{['eq:roughf']}.
  • Figure 5: Solutions of the convection-diffusion problem using the SUPG finite element discretization (VP) and variational inequality (VI) with linear and quadratic Bernstein approximations on a coarse mesh of 5,440 triangles and 2,832 vertices.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 2.1
  • Theorem 3
  • proof