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High-order Virtual Element Method on polyhedral meshes
L. Beirão da Veiga, F. Dassi, A. Russo
TL;DR
The paper addresses the limited three-dimensional numerical validation of VEM beyond the lowest order. It applies a general-order VEM to a diffusion-reaction problem and evaluates convergence, polyhedral meshing, irregular-grid robustness, stabilization, and order dependence, finding the technology practically viable in this setting.
Problem
Three-dimensional VEM numerical experiments had largely been limited to the lowest order, k = 1, leaving general-order performance to be validated.
Method
The study applies a three-dimensional general-order VEM to a diffusion-reaction model and evaluates h-convergence, polyhedral Voronoi meshes, irregular grids, stabilization, and convergence in k.
Results
The experiments achieve the theoretical convergence rate for all tested VEM approximation degrees and mesh types, while convergence in k is exponential with a slight bend at k = 5.
Takeaways & Limitations
The study supports the practical viability of general-order VEM for three-dimensional diffusion-reaction problems on polyhedral meshes, including irregular grids.
Takeaways & Limitations
The analysis assumes meshes whose elements and faces are uniformly star-shaped and whose edges have lengths bounded below proportionally to element diameters.
Abstract
from arXiv · showhide
We develop a numerical assessment of the Virtual Element Method for the discretization of a diffusion-reaction model problem, for higher "polynomial" order k and three space dimensions. Although the main focus of the present study is to illustrate some h-convergence tests for different orders k, we also hint on other interesting aspects such as structured polyhedral Voronoi meshing, robustness in the presence of irregular grids, sensibility to the stabilization parameter and convergence with respect to the order k.
1. Introduction
The paper addresses the limited availability of three-dimensional numerical experiments for VEM beyond the lowest order by assessing general-order schemes on polyhedral meshes. It validates the approach through h-convergence tests and additional studies of meshing, irregularity, stabilization, and polynomial order.
- Three-dimensional VEM experiments in the literature had been limited to the lowest order, k = 1.
- The study numerically validates VEM of general order k for three-dimensional diffusion-reaction problems.
- Standard h-convergence tests assess the method for different approximation orders k.
- Additional tests examine structured polyhedral Voronoi meshes, robustness on irregular grids, sensitivity to stabilization, and convergence with respect to k.
- The paper briefly introduces the three-dimensional model and VEM before presenting an array of numerical tests.
2. The Virtual Element discretization
The section defines a three-dimensional Virtual Element discretization for diffusion-reaction problems on polyhedral meshes, using computable projections and degrees of freedom rather than explicit basis functions. It states consistency, stability, convergence under mesh regularity assumptions, and robustness on irregular meshes.
- Virtual spaces: The method uses a three-dimensional virtual element space on conforming polyhedral partitions, with polynomial degree k and polygonal face spaces.The global space is assembled from local spaces whose face functions glue continuously across edges.
- Computable projections: The degrees of freedom determine computable elliptic and L2 projections onto polynomial spaces without explicitly constructing basis functions.Integration by parts and the defined operators provide the information needed to compute these projections.
- Discrete problem: The discrete bilinear forms combine projection-based consistency with stabilization built from local degree-of-freedom evaluations.The resulting forms are described as consistent and stable, and the discrete problem uses projected load and boundary data.
- Convergence: For meshes whose elements and faces are uniformly star-shaped and whose edges have lengths proportional to element diameter, the method admits an h-dependent convergence estimate for 2 ≤ s ≤ k + 1.The estimate has a constant independent of mesh size and extends to 1 ≤ s < 2 with modified data regularity; an additional result applies on convex or regular domains.
- Scope and robustness: The paper notes that broader mesh assumptions may be possible and reports that numerical tests examine robustness on quite irregular meshes.The stated theoretical result itself is tied to the listed mesh assumptions.
3. Numerical tests
The numerical section evaluates the reliability and robustness of the three-dimensional Virtual Element Method. Its experiments address the method's behavior in the reported numerical setting.
- Numerical evaluation: The section collects numerical results for evaluating the reliability of the Virtual Element Method in three dimensions.
- Numerical evaluation: The numerical results also evaluate the robustness of the Virtual Element Method in three dimensions.
- Numerical evaluation: The tests concern the three-dimensional formulation introduced for the diffusion-reaction setting.
3.1. Meshes and error estimators
The study defines the tested domains, polyhedral mesh families, error measures, and averaged mesh-size parameter used for three-dimensional VEM experiments.
- The experiments use the cube [0, 1]3 and a truncated octahedron as computational domains.
- Three Voronoi mesh types are considered: randomly displaced control points, centroidal Voronoi tessellations, and structured meshes.Structured meshes combine interior cubes with arbitrarily shaped boundary elements.
- Mesh sequences with decreasing size are generated for each mesh type to analyze error convergence rates.
- The errors are evaluated using local degree-k projectors and quantities associated with the VEM discrete solution.
- The approximate L∞ error is computed over selected vertices and internal edge nodes rather than by maximizing over the entire domain.This permits direct computation without projection operators.
- The mesh-size parameter h is measured in an averaged sense using the number of polyhedrons in the mesh.
3.2. Test case 1: h-analysis for diffusion problem on a cube
The cube diffusion test compares Random, CVT, and Structured discretizations across VEM orders, finding the theoretical convergence rate for every order and mesh type.
- The first test solves a diffusion problem on the cube [0, 1]3 with mixed Dirichlet and homogeneous Neumann boundary conditions.
- All three discretizations—Random, CVT, and Structured—are tested, with Structured becoming a standard structured cubic mesh on the cube.
- The convergence data achieve the theoretical rate for all VEM approximation degrees and each mesh type.The rates are reported in Tables 1 and 2 for the H1-seminorm and L2-norm tests.
3.3. Test case 2: h-analysis for diffusion-reaction problem on a polyhedron
The polyhedral diffusion-reaction test examines VEM orders 1–3 on a truncated octahedron using three mesh families. Both error norms follow the expected convergence behavior, while mesh shape modestly affects error magnitude.
- The second test solves a fully Dirichlet diffusion-reaction problem on a truncated octahedron.
- VEM approximation degrees from 1 to 3 are compared across Random, CVT, and Structured meshes.
- For fixed order and mesh family, h is expected to scale as N_dof^-1/3, and both error norms behave as predicted by theory.
- Random meshes produce consistently larger errors than more regular meshes, while Structured meshes give the best results, including against CVT meshes.
3.4. Test case 3: convergence analysis with different k
The study examines convergence as the VEM approximation degree k increases on a fixed truncated-octahedron mesh, comparing stabilization choices. Errors show exponential convergence in k, but the original stabilization bends at k=5 and the diagonal recipe improves the behavior.
- Convergence with respect to k: The fixed truncated-octahedron mesh is used to study convergence for approximation degrees k from 1 to 5.The mesh is a CVT mesh composed of 116 polyhedrons.
- Convergence with respect to k: Both the H1-seminorm and L∞-norm errors exhibit exponential convergence with respect to k.This trend is aligned with existing two-dimensional theory.
- Stabilization behavior: At k = 5, both error curves show a slight bend, attributed to the stabilizing matrix used in the computation.The authors state that the stabilization should be better devised for a spectral approximation strategy.
- Stabilization behavior: The proposed diagonal recipe stabilization cures the observed problem in the present context.The diagonal term uses information already computed in the consistency matrix.
- Stabilization behavior: The diagonal recipe stabilization yields better results than the original stabilization according to the reported slope data.The comparison is presented in Figure 7 and Table 3.
3.5. Test case 4: patch test
The patch test evaluates VEM degrees k from 1 to 5 using polynomial exact solutions on a fixed coarse mesh. Errors remain close to machine precision, although they increase at higher degrees because the involved matrices become more ill-conditioned.
- Patch-test result: VEM recovers polynomial solutions up to machine precision when the solution degree matches the approximation degree.This is the stated patch-test property for the method.
- Test setup: The patch test uses polynomial exact solutions with VEM approximation degrees k from 1 up to 5.The boundary data and right-hand side are chosen consistently with u(x, y, z) := (x + y + z)^k.
- Test setup: The experiments use the same coarse CVT mesh of the truncated octahedron, composed by 116 polyhedrons.The mesh size is not varied because the test focuses on the approximation degree.
- Patch-test result: The measured errors are close to machine precision but become larger for higher VEM approximation degrees.The increase is reported in Table 4.
- Patch-test result: The higher-degree error increase stems from worsening conditioning of the matrices involved in computing the VEM solution.The paper notes that this behavior also occurs in standard FEM.
3.6. Test case 5: stabilizing parameter analysis
The study tests how the stabilization parameter τ affects H1 and L∞ errors for VEM degrees k = 1, 2, 3, and 4. Errors increase for very small or very large τ, while the method remains generally robust across the tested range.
- Parameter study: The analysis varies τ over one hundred logarithmically distributed values from 10^-2 to 10^2 on a fixed CVT mesh of 1024 polyhedrons.Errors are computed in the H1-seminorm and L∞-norm for each corresponding VEM solution.
- Parameter study: Figure 8 compares H1 and L∞ errors as functions of τ for approximation degrees k = 1, 2, 3, and 4.The parameter is shown on a logarithmic scale.
- Observed behavior: Errors grow when τ is chosen very small or very large, with the same qualitative trend for each tested k.The L∞-norm appears more sensitive than the H1-seminorm.
- Observed behavior: The method appears generally robust with respect to the stabilization parameter choice.This conclusion is based on the error trends across the tested approximation degrees.
- Observed behavior: The reported error ratios measure maximum-to-minimum error over τ ∈ [10^-1, 10], spanning a factor of 100 in τ.Table 5 reports these ratios for different approximation degrees.
- Observed behavior: All error ratios in the tested τ range are within an acceptable range.This is the interpretation given for the quantities reported in Table 5.