Ill-conditioning in the Virtual Element Method: stabilizations and bases

1 Introduction

• VEM permits polygonal meshes and high-order methods, but its local spaces include functions that are not explicitly known, so discrete forms are computed through degrees of freedom.Source: §1 · p. 1
• VEM stiffness matrices can be ill-conditioned for high-order methods and on badly shaped polygons, including nonuniformly star-shaped polygons with small edges [13].Source: §1 · p. 1
• Two potential influences on ill-conditioning are the approximate discrete bilinear form, especially its stabilization, and the choice of basis; stabilization effects are noted for 3D VEM [32].Source: §1 · pp. 1–2
• The paper examines whether choices of discrete bilinear forms and VE bases can address ill-conditioning in high-order VEM or on badly shaped polygons.Source: §1 · p. 2
• A proper basis can improve stiffness-matrix conditioning, whereas stabilization choice has little impact in the 2D setting studied.For low-to-moderate order VEM on shape-regular meshes, the standard method remains preferable because its implementation is simpler.Source: §1 · p. 2

2.1 A family of VEM

• The local VEM space combines Pp(K) with virtual functions, and its computable stiffness form uses stabilization on the projector kernel while retaining p-consistency and stability.The polynomial basis used for internal moments does not change the space definition, but it can affect the stiffness-matrix condition number.Source: §2.1 · pp. 3–5

2.2 Choices for the stabilization

• Four literature-based stabilizations are considered; in 2D the standard choice is easy to implement and scales like the H1 seminorm, while observed stability constants vary mildly with p.A different stabilization was reported to improve high-p behavior for a 3D Poisson problem [32].Source: §2.2 · p. 6

2.3 Choices for the basis

• The standard polynomial basis is computationally convenient but harms stiffness-matrix conditioning for high local degrees p and bad-shaped polygons.The basis choice changes the internal VEM bubble functions through the moments used to define them.Source: §2.3 · p. 7
• Three bases dual to internal moments are presented: the standard basis is easy to implement but may cause high stiffness-matrix conditioning, while two orthonormalized alternatives are expected to perform better.The new basis's implementation is not detailed, but is stated to be similar to the second basis's implementation in Appendix A.Source: §2.3 · pp. 7–8
• Using L2(K)-orthonormal polynomials for internal moments partially orthonormalizes the local canonical basis, because VEM spaces also contain nonpolynomial functions.The passage motivates this rationale by noting that the nonpolynomial subspace is asymptotically smaller than the polynomial subspace as the degree of accuracy increases.Source: §2.3 · p. 8

2.4 Stabilizations and bases: the effects on the method

• Stabilization choices affect convergence through the pollution factor α*(p) and stiffness-matrix conditioning; canonical-basis choices change matrix entries and the stabilization, which can also alter α*(p).Applying the same stabilization with different canonical bases can yield different values because the internal degrees of freedom depend on the basis.Source: §2.4 · pp. 8–9

3 Numerical results

• The experiments target conditioning in the p-version and in sequences of badly shaped polygons, including cases that violate the stated geometric assumptions.The condition number is computed as the ratio of the largest to smallest nonzero stiffness-matrix eigenvalue.Source: §3 · p. 9, §3.2 · p. 11, §3.3 · p. 12

3.1 Numerical results: the p version of VEM

• With SK1 fixed, the standard basis has exponential condition-number growth with p, while the other two bases suggest algebraic growth on all three meshes.The experiments also report that the standard basis produces large high-degree errors and can impair the linear solver; the authors recommend one of the alternative bases for high-order VEM.Source: §3.1 · pp. 9–10
• In the p-version tests, the three alternative bases yield algebraic growth, whereas the standard basis yields exponential growth in the condition number.The p-version comparisons use three meshes and hold SK1 fixed.Source: §3.1 · p. 9, Figure 2 · p. 10
• For the p-version error tests, floating-point error for a polynomial exact solution grows with the stiffness-matrix condition number, and the standard basis has a large high-degree error.The three meshes and bases are compared with SK1 fixed.Source: §3.1 · pp. 10–11, Figure 4 · p. 11

3.2 Numerical results: collapsing polygons

• For collapsing polygons, the standard basis leads to dramatic condition-number growth, while the basis obtained through orthonormalization is the safest tested choice.The study also reports no evident stabilization effect in the presented tests.Source: §3.2 · p. 12

3.3 Numerical results: hanging nodes

• For the hanging-node sequence, condition numbers are almost independent of its parameter for the canonical bases, although the standard basis gives higher values.The tested stabilizations again produce basically the same behavior.Source: §3.3 · p. 13

• The study addresses ill-conditioning arising at high polynomial degree and in the presence of highly anisotropic elements.Source: §4 · pp. 13–14
• The tested stabilizations have almost the same effect on condition number, whereas the choice of internal degrees of freedom has a deep impact.Source: §4 · p. 14
• Two practical modifications of the internal degrees of freedom greatly improve high-order VEM and VEM on bad-shaped polygons.Source: §4 · p. 14
• The conclusions are limited to a simple 2D Poisson problem; for 3D problems, stabilization (14) plays a major role, as shown in [15] [32].Source: §4 · p. 14

• The polynomial basis {q2} is obtained by L2 orthonormalization, with stable Gram-Schmidt used in the construction.Source: A Appendix: a hitchhikers guide for… · p. 14
• The appendix notes that implementation details for basis {q1 were discussed in.Source: A Appendix: a hitchhikers guide for… · p. 15
• The local stiffness matrix uses the selected stabilization SK and the projector Π∇_p acting from Vp(K) to Pp(K).Source: A Appendix: a hitchhikers guide for… · p. 16
• The appendix details construction of the local stiffness matrix using the new canonical basis associated with polynomial basis {q2.Source: A Appendix: a hitchhikers guide for… · pp. 14, 16
• The appendix gives the matrix transformation eG = GS · eG · GST, where GS contains the orthonormalizing coefficients.Source: A Appendix: a hitchhikers guide for… · pp. 15–16
• The construction computes internal-degree-of-freedom entries using the L2 orthonormality of the q2 basis.Source: A Appendix: a hitchhikers guide for… · p. 17