Shifted Boundary Method (SBM)

• Shifted Boundary Method (SBM) is an unfitted finite element technique that shifts boundary conditions from curved physical boundaries to nearby surrogate boundaries.
• It employs high-order polynomial corrections to restore optimal convergence rates and simplify simulations without the need for curved meshes.
• SBM integrates seamlessly with existing finite element and finite volume codes, extending its applicability to multiphysics problems, compressible flows, and shock simulations.

The Shifted Boundary Method (SBM) is an unfitted finite element and finite volume technique designed to treat boundary value problems for partial differential equations (PDEs) on complex geometries. SBM "shifts" boundary data from the true, generally curved domain boundary to a nearby surrogate boundary that conforms to an underlying mesh, typically linear, Cartesian, or octree. Boundary conditions at mesh-aligned surrogate boundaries are corrected using high-order polynomial extrapolation based on the solution within the mesh cell, thereby eliminating the need for explicit high-order Taylor expansions or integration over cut cells. This approach delivers high-order accuracy without requiring curved meshes and is compatible with existing finite element and finite volume codes for both continuous and discontinuous Galerkin discretizations. The method is applicable to compressible and incompressible flows, diffusion, elasticity, and multiphysics, including problems with shocks.

1. Geometric Formulation and Closest-Point Maps

SBM operates by replacing the true physical domain $\Omega$ with a surrogate domain $\Omega_h$ composed of mesh cells fully interior to $\Omega$. The surrogate boundary $\Gamma_h = \partial \Omega_h$ generally does not coincide with the true boundary $\Gamma = \partial \Omega$; for each quadrature point $\hat{x} \in \Gamma_h$, the signed distance $d(\hat{x}) = \mathrm{dist}(\hat{x}, \Gamma)$ is computed, where positive values denote interior points. The outward unit normal $n(x)$ is measured at the closest point $x = \mathcal{R}(\hat{x})$ on $\Gamma$, with the closest-point map $\mathcal{R}: \Gamma_h \to \Gamma$ defined as $x = \hat{x} + d(\hat{x}) n(x)$. Maintaining $n(x) \parallel d(\hat{x})$ and $|d(\hat{x})| = \| d(\hat{x}) \|$ is essential for consistency.

2. Polynomial Correction of Boundary Conditions

For high-order consistency, SBM introduces a correction based on extrapolating the cell-local solution polynomial to the true boundary. On each boundary cell $K \in \mathcal{T}_h$, the degree-$p$ polynomial approximation is

$$u_h(\xi) = \sum_{i=1}^D U_i \psi_i(\xi), \quad \xi \in K,$$

where $\{ \psi_i \}$ is a basis for $P_p(K)$.

Rather than explicitly computing derivatives for a Taylor expansion,

$$u(x) = u(\hat{x}) + \nabla u(\hat{x}) \cdot d + \tfrac{1}{2} d^T H(u(\hat{x})) d + \dots,$$

SBM relies on polynomial extrapolation:

$$u_{SBM}(\hat{x}) = u_h(\hat{x}) + [ u_D(x) - u_h(x) ] = u_h(\hat{x}) + \delta u_h(\hat{x}),$$

with $\delta u_h(\hat{x}) := u_D(x) - u_h(x)$, $x = \hat{x} + d n$. This correction is exact for degree-$p$ polynomials, requiring only one additional local evaluation per face quadrature point.

3. Numerical Boundary Condition Enforcement

SBM replaces standard discrete boundary states—such as ghost states in finite volume and DG methods—with shifted, corrected states at surrogate boundary points:

• Far-field (Dirichlet): $U^{bc} = U_h(\hat{x}) + [ U_D(x) - U_h(x) ]$, applied at each quadrature point $\hat{x} \in \Gamma_h$.
• Characteristic far-field: $U_D(x)$ is constructed from prescribed Riemann invariants, evaluated at the projected $x$.
• Slip-wall: For $u \cdot n = w$ on $\Gamma$, the surrogate normal $\tilde{n}$ at $\hat{x}$ is decomposed into directions relative to $n(x)$; only the normal component is corrected:

  $$w_{SBM} = u_h(\hat{x}) \cdot n + [ w - u_h(x) \cdot n ].$$

The ghost state $U^{bc}$ combines the density and tangential velocity from $U_h(\hat{x})$, the corrected normal velocity $w_{SBM}$, and consistent enthalpy values.

The corrected ghost state $U^{bc}$ enters the numerical flux $\hat{H}(U_h^{-}, U^{bc})$ in the boundary DG/FV computations.

4. Algorithmic Implementation

Implementing SBM on a linear mesh with precomputed $d(\hat{x})$ and $n(x)$ entails:

1. Identifying all boundary faces and quadrature points $\hat{x}$ on $\Gamma_h$.
2. For each $\hat{x}$, compute projected $x = \hat{x} + d(\hat{x}) n(x)$.
3. Evaluate the cell polynomial $U_h$ at $x$.
4. Compute the correction $\delta U = U_D(x) - U_h(x)$.
5. Form the corrected boundary/ghost state $U^{bc} = U_h(\hat{x}) + \delta U$.
6. Use $U^{bc}$ in the boundary flux operator.

The additional computational expense is strictly one local polynomial evaluation per boundary quadrature point; required data structures need only store $d$ and $n$ at the quadrature points.

5. Theoretical Analysis and Convergence Rates

Formal analysis (Ciallella et al., 2022) shows that with $\mathcal{O}(h^2)$ boundary approximation (i.e., distance between $\Gamma$ and $\Gamma_h$ scales as $h^2$) and degree-$p$ accuracy in the interior, the SBM polynomial shift restores optimal convergence:

$$\| u - u_h \|_{L_2(\Omega)} = \mathcal{O}(h^{p+1}).$$

Numerical rates for Dirichlet BCs on 2D curved domains:

• DG–P2 without SBM: rate $\approx 2$, with SBM: rate $\approx 3$.
• DG–P3 without SBM: rate $\approx 2$, with SBM: rate $\approx 4$.

In analogous 3D tests (spheres), DG–P3 with SBM achieves $O(h^4)$ L₂ convergence.

6. Extension to Shocked Flows

SBM boundary correction applies unchanged in time-dependent ADER-DG frameworks with MOOD limiters for flows with shocks. Boundary cells flagged as troubled and downgraded to subcell FV still use the shifted Dirichlet or slip-wall state for fluxes. For slip-wall treatments, a Rusanov-type penalty term can be added for stability:

$\mathcal{P}_w = \alpha_w (U_h(\hat{x}) - U^{bc}),$

without compromising accuracy.

7. Practical Significance and Applicability

The SBM polynomial correction method is compatible with standard finite element and finite volume codes, demanding only minimal extension for data storage of $d$ and $n$, and a negligible additional computational cost for boundary point evaluation. By eliminating the requirement for curved meshes and explicit high-order Taylor expansions, SBM streamlines high-order simulations of compressible flows in domains with curved boundaries. It yields high-order boundary consistency and optimal convergence for all types of Euler boundary conditions—Dirichlet, characteristic far-field, and slip-wall. This is especially beneficial for problems involving curvature, shocks, and complex geometries (Ciallella et al., 2022).

The SBM has been adopted and validated for a broad class of PDEs, including Poisson, elasticity, Navier–Stokes, and hyperbolic conservation laws. Its formulation supports efficient matrix-free and parallel implementations, robust scaling under geometric complexity, and seamless extension to immersed and adaptive mesh frameworks.

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Key reference:

"Shifted boundary polynomial corrections for compressible flows: high order on curved domains using linear meshes" (Ciallella et al., 2022)