Source-linked AI summary
The non-conforming virtual element method for the Stokes equations
Andrea Cangiani, Vitaliy Gyrya, Gianmarco Manzini
TL;DR
The paper addresses how to construct a non-conforming VEM for steady Stokes velocity and pressure on broad polygonal and polyhedral mesh families. It uses polynomially containing virtual velocity spaces, discontinuous polynomial pressure, and computable projections, proving stability and error bounds. Numerical results agree with the predicted convergence rates and support high-order approximation.
Problem
Constructing higher-order non-conforming finite element spaces beyond standard element shapes is difficult, especially on general polygonal and polyhedral meshes.
Method
The method uses non-conforming virtual velocity spaces containing degree-k polynomials, discontinuous degree-(k−1) pressure spaces, and computable polynomial projections.
Results
The velocity-pressure spaces satisfy the inf-sup condition, the discrete problem has a unique solution, and numerical error curves agree with the predicted convergence rates.
Takeaways & Limitations
The formulation provides a unified high-order non-conforming VEM for two- and three-dimensional problems on general polygonal and polyhedral meshes.
Takeaways & Limitations
The treatment does not address the L2-projection issue associated with low-order terms because that projection is not needed in this work.
Abstract
from arXiv · showhide
We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.
1. Introduction.
The paper develops a non-conforming VEM for steady Stokes problems on general polygonal and polyhedral meshes. Its formulation combines implicitly defined virtual spaces with polynomial pressure approximation and an inf-sup-stable velocity-pressure discretization.
- 1. Introduction.: The paper situates the method among non-conforming finite elements, VEM formulations, discontinuous Galerkin methods, and other polygonal-mesh approaches.A comparison with these alternative approaches is identified as future work.
- 1. Introduction.: The proposed non-conforming VEM supports arbitrary polynomial degrees on very general polygonal and polyhedral elements in two and three dimensions.This addresses the difficulty of constructing higher-order non-conforming spaces beyond triangular meshes.
- 1. Introduction.: Virtual trial and test functions are defined implicitly through local boundary-value problems and are not explicitly constructed in practice.The VEM framework therefore works with polynomial information and degrees of freedom rather than explicit non-polynomial shape functions.
- 1. Introduction.: Each velocity component uses a non-conforming virtual space containing polynomials through degree k, while pressure uses discontinuous polynomials through degree k−1.The method approximates gradient and divergence through exactly computable polynomial projections.
- 1. Introduction.: The method is designed around an inf-sup-compatible pair of finite element spaces for the Stokes velocity-pressure formulation.The paper positions this stability requirement as central to the proposed discretization.
2. Continuous Stokes problem and discrete formulation.
The discrete formulation approximates the steady Stokes problem on polygonal or polyhedral meshes using discrete velocity and pressure spaces and computable bilinear forms. Its well-posedness is tied to coercivity on the divergence kernel and a discrete inf-sup condition.
- 2. Continuous Stokes problem and discrete formulation.: Well-posedness of the continuous Stokes problem follows from coercivity on the kernel of the divergence form and the inf-sup condition.These same structural properties motivate the analysis of the discrete formulation.
- 2. Continuous Stokes problem and discrete formulation.: The steady Stokes problem is posed in variational form for velocity and pressure on a polygonal or polyhedral domain.The framework applies in two and three spatial dimensions using standard Sobolev-space notation.
- 2. Continuous Stokes problem and discrete formulation.: The VEM discretization introduces finite-dimensional velocity and pressure spaces with discrete counterparts of the continuous bilinear forms.The construction of these spaces and forms is the focus of the subsequent virtual-element framework.
- 2. Continuous Stokes problem and discrete formulation.: The approximate formulation uses suitable polynomial approximations of the forcing and boundary data in the discrete problem.The discrete problem’s well-posedness is established through a discrete inf-sup condition together with coercivity and stability properties.
3. Virtual element framework.
The framework combines polygonal/polyhedral meshes, discontinuous polynomial pressures, and nonconforming virtual velocity spaces whose polynomial projections and degrees of freedom make the method computable without explicitly evaluating virtual functions.
- 3.1. Mesh regularity and polynomial approximation: The mesh supports general polygonal and polyhedral elements under regularity conditions including star-shapedness and uniformly controlled interfaces.These assumptions cover non-convex elements, hanging-node configurations, and both two- and three-dimensional settings.
- 3.2. Discrete pressure space: The pressure space uses discontinuous piecewise polynomials of degree at most k −1 on the mesh partition.
- 3.3. Scalar non-conforming virtual element space: The scalar nonconforming virtual space is characterized by edge/face moments of degree k −1 and internal moments of degree k −2.These degrees of freedom are unisolvent, and their count is NE = νE N_d−1,k−1 + N_d,k−2.
- 3.3. Scalar non-conforming virtual element space: Each local virtual space contains P_k(E) plus non-polynomial functions represented discretely through degrees of freedom rather than explicit evaluation.The virtual component is considered expensive to evaluate, but the discrete representation suffices for constructing the method.
- 3.3. Scalar non-conforming virtual element space: The Ritz-Galerkin projection is computable from the degrees of freedom, while the L2 projector applied to first derivatives is also available for the Stokes formulation.Together these projectors provide the computable polynomial information needed to define the virtual formulation.
- 3.3. Scalar non-conforming virtual element space: The full-order L2 projection of virtual functions is unavailable from the chosen degrees of freedom because it would require internal moments through degree k.The paper therefore does not pursue that projection because it is not needed for the present formulation.
4. Error analysis.
The analysis establishes well-posedness through a mesh-independent inf-sup condition and derives a priori error bounds for velocity and pressure, including non-conformity effects.
- 4.1. Existence and uniqueness: The discrete virtual element problem has a unique solution because coercivity on the discrete divergence-free kernel combines with the proven inf-sup property.The inf-sup constant is strictly positive and independent of h.
- 4.3. Error estimate for the velocity: The H1 velocity error bound combines source approximation, conformity, consistency, stability, and non-conformity contributions.The non-conformity contribution is nonzero because the discrete velocity space is non-conforming.
- 4.3. Error estimate for the velocity: For sufficiently regular exact velocity and pressure, the velocity approximation admits an optimal-order H1 estimate with a constant independent of h.The constant depends on stability and mesh-regularity constants; optimal L2 velocity estimates can also be derived by duality.
- 4.4. Error estimate for the pressure: The pressure approximation satisfies an L2 a priori error bound whose constant depends on stability, inf-sup, and mesh-regularity constants.The pressure result is stated under the same regularity assumptions used for the velocity estimate.
5. Implementation details.
Implementation reduces the virtual formulation to computable polynomial projections and matrix operations, with consistency and stabilization terms assembled from degrees of freedom.
- 5. Implementation details: Once the projection operators are constructed, local bilinear-form terms are computable as polynomial integrals except for stabilization.The stabilization term is defined directly through the projector and degrees of freedom.
- 5. Implementation details: Projectors involving divergence and derivatives are obtained from corresponding scalar projection formulas or referenced constructions.These projectors support the computable virtual formulation for the Stokes problem.
- 5. Implementation details: The consistency and stability terms of the local velocity bilinear form are assembled using matrix representations of polynomial expansions and projectors.The implementation introduces coefficient matrices for monomial bases and projection operators.
- 5. Implementation details: Numerical tests first verify polynomial consistency on polygonal meshes for manufactured solutions with polynomial degrees m = 1 to 4.The experiments are intended to confirm the preceding a priori analysis.
6. Numerical Results.
The numerical study tests the non-conforming VEM across three mesh families and polynomial orders k=1 to 4, comparing pressure, velocity, and gradient errors with predicted convergence slopes. The reported results agree very well with the theoretical convergence rates.
- Test setup: The method is evaluated for k=1, 2, 3, 4 on three mesh sequences, M1, M2, and M3, using successive mesh refinements.M1 uses perturbed quadrilateral meshes, M2 derives meshes with triangular subdivisions and barycentric connections, and M3 uses another polygonal construction.
- Error evaluation: The pressure and velocity approximations are assessed through computable polynomial quantities and piecewise polynomial fields, respectively.For velocity, the study compares Π_k−1(∇u_h) with the exact velocity and gradient; for pressure, it compares p_h and ∇p_h with their exact counterparts.
- Error evaluation: Figures 6.2 and 6.3 plot relative pressure, velocity, and gradient errors against mesh size h for all three mesh families.Left panels show field errors, right panels show gradient errors; expected slopes are marked directly on the log-log plots.
- Convergence results: The numerical results are in very good agreement with the convergence rates predicted by the analysis.The experiments therefore confirm the theoretical convergence behavior across the reported error curves.
7. Conclusions.
The paper concludes that its non-conforming VEM provides a unified, stable, and optimally convergent approach for steady Stokes problems on general polygonal and polyhedral meshes. Its approximation order is controlled by the polynomial degree k.
- Conclusions: The method constructs arbitrary-order approximations in two and three dimensions on general polygonal and polyhedral meshes.The formulation allows non-convex polygons and polyhedra with parallel adjacent interfaces.
- Conclusions: The velocity space contains degree-k polynomial vectors plus implicit non-polynomial functions, while the pressure space consists of degree-(k−1) polynomials.The non-polynomial velocity functions are handled through degrees of freedom rather than explicitly computed representations.
- Conclusions: The velocity-pressure pair satisfies the inf-sup condition, establishing stability and well-posedness of the discrete scheme.The paper also derives optimal a priori error estimates for both velocity and pressure.
- Conclusions: The approximation accuracy is determined by the polynomial degree k, with optimal convergence proved for the velocity and pressure fields.