Mixed HP -finite element approximations on geometric edge and boundary layer meshes in three dimensions

Summary: In this paper, we consider the Stokes problem in a three-dimensional polyhedral domain discretized with hp finite elements of type Qk for the velocity and Qk-2 for the pressure, defined on hexahedral meshes anisotropically and non quasi-uniformly refined towards faces, edges, and corners. The inf-sup constant of the discretized problem is independent of arbitrarily large aspect ratios. Our work generalizes a recent result for two-dimensional problems in [10, 11

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysi

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mes

Mixed hp‐DGFEM for incompressible flows II: Geometric edge meshes

We consider the Stokes problem of incompressible fluid flow in three‐dimensional polyhedral domains discretized on hexahedral meshes with hp‐discontinuous Galerkin finite elements of type Qk for the velocity and Qk−1 for the pressure. We prove that these elements are inf‐sup stable on geometric edge meshes that are refined anisotropically and non‐quasiuniformly towards edges and corners. The discrete inf‐sup constant is shown to be independent of the aspect ratio of the anisotropic elements and is of O(k−3/2) in the polynomial degree k, as in the case of conforming Qk−Qk−2 approximations on the same meshe

Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM

We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a delta-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on delta. We apply the obtained estimates to show exponential convergence with rate O(exp(−b square root N)), N being the number of degrees of freedom and b>0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(−b cubic root N )), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $\Omega\subset\mathbb{R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for the space-time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable \emph{sparse} space-time version of the DG scheme. The latter scheme is based on the so-called \emph{combination formula}, in conjunction with a family of anisotropic space-time DG-discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\Omega$ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space-time DG schemes.Comment: 38 pages, 8 figure

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $\Omega\subset\mathbb{R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for the space-time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable \emph{sparse} space-time version of the DG scheme. The latter scheme is based on the so-called \emph{combination formula}, in conjunction with a family of anisotropic space-time DG-discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\Omega$ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space-time DG schemes.Comment: 38 pages, 8 figure

Pre-culture of human mesenchymal stromal cells in spheroids facilitates chondrogenesis at a low total cell count upon embedding in biomaterials to generate cartilage microtissues

Material-assisted cartilage tissue engineering has limited application in cartilage treatment due to hypertrophic tissue formation and high cell counts required. This study aimed at investigating the potential of human mesenchymal stromal cell (hMSC) spheroids embedded in biomaterials to study the effect of biomaterial composition on cell differentiation. Pre-cultured (3 days, chondrogenic differentiation media) spheroids (250 cells/spheroid) were embedded in tyramine-modified hyaluronic acid (THA) and collagen type I (Col) composite hydrogels (four combinations of THA (12.5 vs 16.7 mg/ml) and Col (2.5 vs 1.7 mg/ml) content) at a cell density of 5 × 106 cells/ml (2 × 104 spheroids/ml). Macropellets derived from single hMSCs (2.5 × 105 cells, ScMP) or hMSC spheroids (2.5 × 105 cells, 103 spheroids, SpMP) served as control. hMSC differentiation was analyzed using glycosaminoglycan (GAG) quantification, gene expression analysis and (immuno-)histology. Embedding of hMSC spheroids in THA-Col induced chondrogenic differentiation marked by upregulation of aggrecan (ACAN) and COL2A1, and the production of GAGs. Lower THA led to more pronounced chondrogenic phenotype compared to higher THA content. Col content had no significant influence on hMSC chondrogenesis. Pellet cultures showed an upregulation in chondrogenic-associated genes and production of GAGs with less upregulation of hypertrophic-associated genes in SpMP culture compared to ScMP group. This study presents hMSC pre-culture in spheroids as promising approach to study chondrogenic differentiation after biomaterial encapsulation at low total cell count (5 × 106/ml) without compromising chondrogenic matrix production. This approach can be applied to assemble microtissues in biomaterials to generate large cartilage construct. Statement of significance: In vitro studies investigating the chondrogenic potential of biomaterials are limited due to the low cell-cell contact of encapsulated single cells. Here, we introduce the use of pre-cultured hMSC spheroids to study chondrogenesis upon encapsulation in a biomaterial. The use of spheroids takes advantage of the high cell-cell contact within each spheroid being critical in the early chondrogenesis of hMSCs. At a low seeding density of 5·106 cells/ml (2 × 104 spheroids/ml) we demonstrated hMSC chondrogenesis and cartilaginous matrix deposition. Our results indicate that the pre-culture might have a beneficial effect on hypertrophic gene expression without compromising chondrogenic differentiation. This approach has shown potential to assemble microtissues (here spheroids) in biomaterials to generate large cartilage constructs and to study the effect of biomaterial composition on cell alignment and migration.</p

Loose Bodies Found in the Human Intra-Articular Space Showed Characteristics Similar to Endochondral Bone Formation

Objective: Loose bodies are free-floating tissues of cartilage and bone that can cause pain, swelling, the inability to straighten the knee, or intermittent locking of the knee. Loose bodies can arise from degenerative joint disease, flake fractures, osteochondritis dissecans, or chondromatosis. We hypothesized that loose bodies can be classified in stages with tissue characteristics similar to endochondral ossification.Design: Loose bodies were harvested from patients undergoing joint replacement. Samples were processed for histology, gene expression analysis, and micro-computed tomography (µCT). Cartilage- and bone-related genes and proteins were selected for immunofluorescence stainings (collagen type I, II, and X, SOX9 [SRY-box transcription factor 9], and MMP13 [matrix metalloproteinase 13]) and gene expression analysis (FN [fibronectin], COL1A1, COL2A1, COL10A1, SOX9, MMP13, and aggrecan [ACAN]). Results: Loose bodies were grouped in 4 stages: fibrous, (mineralized) cartilaginous, cartilage and bone, and bone. Hyaline-like cartilage tissue with Benninghoff arcades was present in stages 2 and 3. A transition from cartilaginous to mineralized tissue and bone trabecula was defined by an increase in COL1A1 and COL10A1 (stage 3 vs. 4: p = 0.047) positive area. Stage 4 showed typical trabecular bone tissue. The relative volume of calcified tissue (mineralized cartilage and bone tissue) decreased with stages (stages 1-2 vs. 3: p = 0.002; stage 1-2 vs. 4: p = 0.012). COL2A1 expression and stained area decreased from stages 1-2 to 4 (p = 0.010 and p = 0.004). ACAN expression decreased from stage 1-2 to stage 3 (p = 0.049) and stage 4 (p = 0.002). Conclusion: Loose bodies show tissue characteristics similar to endochondral ossification. They are probably a relevant substrate for regenerative therapeutic interventions in joint disease.</p