Edge Bubble Function in FEM

• Edge bubble functions are specialized enrichment functions in FEM that vanish on element boundaries to capture fine-scale sub-element behavior.
• They are computed through a least-squares minimization process that optimizes the bubble coefficient for accurate approximation of differential equations.
• Integrating edge bubbles into FEM enables precise resolution of sharp gradients and localized phenomena without increasing global degrees of freedom.

An edge bubble function in the context of finite element methods (FEM) refers to a supplementary enrichment function defined within an individual element, vanishing on its boundaries, and designed to enhance the approximation space for improved resolution of fine-scale, sub-element behavior. These functions are particularly valuable for modeling multi-scale transport phenomena and capturing sharp gradients, boundary layers, or localized effects that standard linear shape functions may fail to resolve efficiently.

1. Mathematical Definition and Construction

A bubble function, denoted here as $B_{2,j}(x)$ for an element $I_j = [x_j, x_{j+1}]$, is constructed so that $B_{2,j}(x_j) = B_{2,j}(x_{j+1}) = 0$ (i.e., it is zero on element boundaries) and nonzero in the interior. In the least-squares enrichment approach, the bubble function augments a standard linear element ansatz: $$u_j(x) = (x_{j+1} - x) u_j + (x - x_j) u_{j+1} + c_j (x_{j+1} - x)(x - x_j)$$ where the last term is the bubble function and $c_j$ is an undetermined coefficient.

The optimal $c_j$ is determined for a given PDE (e.g., $\epsilon u'' + k u' + u = f$) by minimizing the squared residual of the equation over the element: $J_B = \int_{x_j}^{x_{j+1}} R_B^2(x) \, dx$ with $R_B(x)$ the residual after inserting $u_j(x)$ into the PDE.

The minimization $\frac{\partial J_B}{\partial c_j} = 0$ yields $c_j$ as a linear function of the nodal values, with explicit form derived in the paper for common transport equations.

2. Integration into the Finite Element Framework

The enriched trial functions are locally constructed for each element and used in assembling the global stiffness matrix. Importantly, the addition of bubble functions does not increase the number of global degrees of freedom, as the bubble coefficients are condensed/eliminated at the element level. Implementation steps:

• For each element, augment the standard shape functions with the polynomial bubble and compute the local stiffness matrix using the enriched basis.
• Determine $c_j$ via least squares minimization of the element residual.
• Globally assemble as usual; the static condensation ensures system size is unchanged.

For multidimensional problems (e.g., on rectangles or quadrilaterals), tensor-product or multivariate polynomial bubbles such as $B(x,y) = c\,x(1-x)y(1-y)$ are used, with coefficients determined analogously via minimization of the multi-dimensional residual.

3. Accuracy Improvements and Computational Impact

Bubble enrichment enables the approximation of sharp internal or boundary layers, and multiscale features without the need for mesh refinement, achieving high accuracy even on relatively coarse grids. For linear transport or reaction-diffusion problems, enrichment with a quadratic or cubic bubble function can bring the FEM solution significantly closer to the analytic solution compared to the standard linear elements, as substantiated by numerical results (see Figure 1 and Tables 1–2 in the paper).

No additional global degrees of freedom are incurred, and the computational cost remains close to standard FEM assembly, since all bubble computations are performed locally and condensed out.

Feature	Standard Linear FEM	Bubble-Enriched FEM
Mesh requirement	Fine mesh	Coarse mesh works
Degrees of freedom	Grows with mesh	No increase
Sub-element features	Poorly captured	Well captured
Automation	Direct	Direct (least squares)
Extension to multi-D	Easy	Algebraically more complex, feasible
High gradient problems	Requires refinement	Possible with higher-order bubbles

4. Case Studies and Demonstrations

• Steady-state reaction-diffusion: Bubble-enriched FEM captures boundary layers and multiscale features accurately on coarse meshes.
• Transient problems: Enriched elements yield solutions nearly matching analytic results with minimal mesh, outperforming standard elements in both transient and steady cases.

5. Higher-Order Bubbles and Mesh Refinement

For strongly convection- or reaction-dominated regimes (e.g., problems with high Péclet or Damköhler numbers), quadratic bubble enrichment may not suffice, and higher polynomial order (cubic, quartic, etc.) bubbles or actual mesh refinement may be required for resolving extremely thin layers. The need for higher-order enrichment is signaled by the scale separation between the feature of interest and the mesh size.

6. Multidimensional Generalization

The method generalizes to higher dimensions by either taking tensor products of one-dimensional bubble polynomials or formulating a least-squares minimization for multivariate bubble forms. For example, on a square: $u(x, y) = (1-x)u_0 + x u_1 + c\,x(1-x)y(1-y)$ with $c$ determined by minimizing the squared residual of the governing equation over the element.

7. Summary and Implementation Considerations

• Edge bubble functions provide a localized, polynomial enrichment to FEM elements, allowing for accurate sub-element resolution of strong gradients and localized features.
• The least-squares approach to determining bubble coefficients ensures the methodology can be automated and applied to complex or nonlinear PDEs.
• No increase in the size of the linear system is necessary; all bubble-related work is handled at the element level.
• The method is especially effective when gradient scales are comparable to or moderately smaller than the element size; for extreme scale separation, combining higher-order enrichment and mesh refinement is advised.

In conclusion, least-squares-based edge bubble function enrichment offers FEM practitioners a robust, efficient technique for accurate multi-scale transport and reaction problems, significantly extending the reach of standard finite element formulations without increased computational overhead for modest levels of problem stiffness.