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Virtual Element Methods for general second order elliptic problems on polygonal meshes
L. Beirão da Veiga, F. Brezzi, L. D. Marini, A. Russo
TL;DR
General variable-coefficient second-order elliptic problems are challenging for VEM because straightforward coefficient extensions can lose optimality. The paper systematically uses computable L2 projections in the VEM discretization, with projected bilinear forms and stabilization. The analysis establishes unique discrete solutions and H1/L2 error estimates, while numerical results show expected behavior and stability with respect to polygon shape.
Problem
VEM extensions from constant to variable coefficients are less trivial, and simple-minded approaches can lose optimality, especially for higher-order methods.
Method
The method systematically makes L2 projections computable from degrees of freedom and uses projected trial and test functions in the VEM discretization.
Results
The analysis proves unique discrete solutions with H1 and L2 error estimates, while numerical results show expected behavior and remarkable stability with respect to polygon shape.
Takeaways & Limitations
Projection-based VEM provides an accurate and robust treatment of the stated general elliptic problems, whereas the earlier simple-minded variable-coefficient extension can be sub-optimal.
Takeaways & Limitations
The method assumes uniformly star-shaped elements with edge lengths bounded below relative to element diameter, and the earlier formulation suffers heavy convergence-order losses for k ≥3.
Abstract
from arXiv · showhide
We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [59] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [9]), we use here, in a systematic way, the L^2-projection operators as designed in [1]. In particular, the present method does not reduce to the original Virtual Element Method of [9] for simpler problems as the classical Laplace operator (apart from the lowest order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [9] to the case of variable coefficients produces, in general, sub-optimal results.
1. Introduction
The paper develops Virtual Element Methods for general two-dimensional second-order elliptic problems, emphasizing computable L2 projections and the difficulty of variable coefficients. It also situates VEM among polygonal discretizations and discusses representing internal values through projected polynomials.
- The paper designs and analyzes VEM for two-dimensional diffusion-convection-reaction problems with variable coefficients.
- For VEM, local trial and test functions solve elementwise PDE problems but are not explicitly solved; polynomial-consistent bilinear-form terms are computed and the remainder stabilized.
- Related polygonal methods may use two distinct polynomial trial and test functions, whereas VEM uses one shared non-polynomial function with boundary and internal degrees of freedom.
- The newer approach makes L2 projections exactly computable from degrees of freedom and systematically uses projected trial and test functions.
- A VEM solution’s internal values can be represented using its L2 projection onto piecewise polynomials of degree k.
2. The problem and the adjoint problem
The problem is posed on a bounded convex polygonal domain for a variable-coefficient diffusion-convection-reaction operator, under smoothness, ellipticity, and regularity assumptions. These assumptions support existence and uniqueness for the problem and its formal adjoint.
- The domain is a bounded convex polygon in R2, with smooth scalar diffusion and reaction coefficients and a smooth vector convection coefficient.
- The diffusion coefficient satisfies the uniform positivity condition κ(x) ≥ κ0 > 0 throughout the domain.
- The differential operator combines diffusion, convection, reaction, and a forcing term through Lp = div(−κ(x)∇p + b(x)p) + γ(x)p = f(x).
- The stated regularity assumptions imply existence and uniqueness for both the original and formal adjoint problems.
- The analysis notes that s-regularity with s > 1 would suffice, although it does not minimize the regularity assumptions.
- Using a full diffusion tensor instead of a scalar coefficient would not substantially change the analysis, and numerical experiments use a full tensor.
3. Variational formulation
The variational formulation uses a bilinear form combining diffusion, convection, and reaction contributions. Under the coefficient assumptions and regularity condition, the formulation has a unique solution even without the stated coercivity premise.
- The variational bilinear form is decomposed as B(p, q) = a(p, q) + b(p, q) + c(p, q).
- The paper formulates the boundary-value problem variationally before introducing its virtual element discretization.
- The coefficient assumptions imply the required properties of the bilinear form B.
- The resulting conditions imply existence and uniqueness of the variational problem’s solution.
4. VEM approximation
The VEM approximation is built on polygonal meshes using degrees of freedom that make polynomial projections computable, then combines projected bilinear forms with stabilization. The formulation is designed for general coefficients but differs from an earlier choice beyond low order.
- The Virtual Element space: The mesh consists of star-shaped polygonal elements, with geometric assumptions controlling element star-shapedness and edge lengths uniformly across decompositions.
- The Virtual Element space: For each element, the degrees of freedom include vertex values, edge moments, and internal moments of the virtual function.
- The Virtual Element space: These degrees of freedom uniquely determine polynomials in Pk(E), enabling construction of computable projection operators.
- The Virtual Element space: The method constructs local projection operators from degrees of freedom and collects them into global projections onto piecewise polynomial spaces.
- The Virtual Element space: The L2 projection, gradient projection, interpolation operator, and related polynomial-space properties are established as computable from the degrees of freedom.
- The discrete problem: The discrete bilinear forms combine projected diffusion terms with a symmetric stabilization and separately define convection, reaction, and loading contributions.
- The discrete problem: The stabilization term is defined only on the virtual element space, distinguishing the discrete form from the corresponding continuous form.
- The discrete problem: The earlier choice coincides with the present one for k = 1 but differs for k ≥2, with heavy convergence-order losses reported for k ≥3.
5. Error estimates
This section develops approximation, continuity, consistency, and stability estimates for the proposed Virtual Element method, culminating in H1 and L2 error results for sufficiently small h.
- The section derives error estimates for the proposed method using approximation results based on polygonal-mesh regularity assumptions.The analysis begins with preliminary approximation and continuity lemmas.
- The discrete bilinear form Bh is shown to be continuous, with a constant depending on κ, b, and γ but independent of h.
- The consistency analysis bounds projection and coefficient-approximation errors, including an O(h^k) estimate for κ∇Π0^k p.For interpolated functions, the consistency estimate can gain an extra power of h.
- A discrete stability lemma supports existence and uniqueness of the discrete problem for sufficiently small h.The proof uses the discrete counterpart of the continuous stability condition and approximation estimates.
- H1 Estimate: For sufficiently small h, problem (4.17) has a unique solution ph ∈ Qk_h and satisfies the stated H1 error estimate, with a constant independent of h.The constant depends on κ, β, and γ.
- L2 estimate: Theorem 5.12 establishes the corresponding L2 error estimate for sufficiently small h, with a constant depending on κ, β, and γ but independent of h.The proof uses duality arguments and the auxiliary problem's regularity.
- The analysis notes that the three-dimensional extension should not present major difficulties, but omits it to avoid heavier notation and technicalities.
6. Numerical Experiments
The experiments test VEM on several polygonal mesh sequences using relative L2, H1, and pointwise errors for polynomial degrees k = 1 and k = 4. The method shows convergence and stability across polygon shapes, while the simple-minded extension (4.18) loses convergence order for k = 4.
- Experimental setup: The test problem is posed on the unit square with an exact solution, and errors are evaluated through the L2-projection of the VEM solution.The experiments compare the exact solution with the projection of the discrete solution because the VEM solution is not explicitly known inside elements.
- Experimental setup: Four mesh sequences comprise random Voronoi, centroidal Voronoi, square, and highly non-convex polygonal meshes.The sequences use 25, 100, 400, and 1600 polygons or squares, with the concave sequence formed by subdividing squares into non-convex polygons.
- Case k = 1: For k = 1, convergence is examined using relative L2, H1, pointwise, and alternative-method L2 errors across the four mesh sequences.The alternative method is (4.18), and its k = 1 L2 graphs are identical to those of the main method.
- Case k = 4: For k = 4, the experiments again measure relative L2, H1, and pointwise errors, alongside the L2 error from method (4.18).The reported comparison states that method (4.18) produces a heavy loss in the order of convergence.
- Conclusions: The VEM behaves as expected and remains remarkably stable with respect to the shapes of the mesh polygons.This conclusion covers the polygonal mesh families used in the experiments, including the non-convex sequence.