The proximal Galerkin method for non-symmetric variational inequalities

Abstract

We introduce the proximal Galerkin (PG) method for non-symmetric variational inequalities. The proposed approach is asymptotically mesh-independent and yields constraint-preserving approximations. We present both a conforming PG formulation and a hybrid mixed first-order system variant (FOSPG). We establish optimal a priori error estimates for each variant, which are verified numerically. We conclude by applying the method to American option pricing, free boundary problems in porous media, advection-diffusion with a semipermeable boundary, and the enforcement of discrete maximum principles.

Figures (7)

• Figure 1: Two--dimensional dam with a left sloping wall. The extended domain $\Omega=OABC$ contains the unknown saturated region, whose free surface $\varphi(x)$ satisfies $h_l\ge \varphi(x)\ge h_r$.
• Figure 2: Convergence history of the proximal iterates $u_h^k$ and averages $\overline{u}_h^k$ for the circular obstacle problem (\ref{['eq:circular_sol']} in \ref{['sec:conv']}) with step size $\alpha_k=2^{k-1}$ and initial mesh size $h_0=1/16$. Left: The Cauchy error, measured by the successive differences of the iterates. Center: The optimization error relative to the converged discrete solution $u_h$. Right: The total error $|u - u_h^k|_{H^1(\mathcal{T}_h)}$.
• Figure 3: Convergence history of the proximal iterates $u_h^k$ and averages $\overline{u}_h^k$ for the biactive problem (\ref{['eq:biactive']} in \ref{['sec:conv']}) with step size $\alpha_k=2^{k-1}$. Left: The Cauchy error, measured by the successive differences of the iterates. Center: The optimization error relative to the converged discrete solution $u_h$. Right: The total error $|u - u_h^k|_{H^1(\mathcal{T}_h)}$.
• Figure 4: Left: The converged solution $\nabla \mathcal{R}^*(\psi_h)$ for the option pricing example at the final time $T = 0.25$. Right: The estimated active set at $T=0.25$.
• Figure 5: The potential $u$ and contour lines at $u=0.96,0.98$, and $1$ with various $\phi$ for the semi-permeable membrane example \ref{['sec:semi_permeable']}. Depending on $\phi$, the active set is empty ($\phi=0.96$, left), a proper subset of $\Gamma_S$ ($\phi=0.98$, center), or $\Gamma_S$ ($\phi=1$, right), respectively.

Theorems & Definitions (33)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Example 2.1: Advection-diffusion problems with bound constraints
• Remark 1: Discrete Maximum Principle
• Example 2.2: American option pricing
• Example 2.3: Semi-permeable boundary conditions
• Example 2.4: Free boundary problem in porous medium flow
• Theorem 3.1