Simple Interpolation Research Articles

Application of Resistance Ring Array Sensors for Oil–Water Two-Phase Flow Water Holdup Imaging in Horizontal Wells

Unconventional oil and gas reservoirs are frequently developed using inclined and horizontal wells, leading to intricate multiphase flow patterns due to spatial asymmetry surrounding the wellbore and gravitational differentiation effects. Through the examination of water holdup imaging, the spatial arrangement of oil and water phases within the wellbore may be clearly depicted, yielding critical information for precisely assessing the ratios of oil and gas. This study employed No. 10 industrial white oil and tap water as fluid media, with measurements obtained using a resistive ring array tool (RAT) to evaluate its response properties over the wellbore cross-section. The data gathered throughout the trials were analyzed by two-dimensional interpolation imaging utilizing 2020 version MATLAB software. To enhance the analysis of water holdup distribution in the wellbore, three interpolation algorithms were utilized: Simple Linear Interpolation (SLI), Inverse Distance Weighting Interpolation (IDWI), and Ordinary Kriging Interpolation (OKI). The results indicated that RAT operates effectively in medium and low flow circumstances, correctly representing the real distribution of oil and water phases while yielding more dependable water holdup data. The SLI algorithm effectively delineates the oil-water interface during stratified flow of oil and water phases, rendering it the optimal algorithm for determining water holdup in standard flow patterns. Under DW/O&W and DO/W&W flow patterns, SLI continues to perform well; however, the accuracy of IDWI and OKI markedly enhances, with IDWI more effectively delineating the attributes of intricate mixed flow and more precisely representing the dynamic fluid distribution. Under DW/O and DO/W flow patterns, the OKI algorithm exhibits optimal performance in these intricate dispersed flow patterns. OKI more precisely represents the dynamic distribution of dispersed oil and water due to its capacity to simulate the spatial correlation of both phases, surpassing both SLI and IDWI.

Application of Imaging Algorithms for Gas-Water Two-Phase Array Fiber Holdup Meters in Horizontal Wells.

The flow dynamics of low-yield horizontal wells demonstrate considerable complexity and unpredictability, chiefly attributable to the structural attributes of the wellbore and the interplay of gas-water two-phase flow. In horizontal wellbores, precisely predicting flow patterns using conventional approaches is often problematic. Consequently, accurate monitoring and analysis of water holdup in gas-water two-phase flows are essential. This study performs a gas-water two-phase flow simulation experiment under diverse total flow and water cut conditions, utilizing air and tap water to represent downhole gas and formation water, respectively. The experiment relies on the measurement principles of an array fiber holdup meter (GAT) and the response characteristics of the sensors. In the experiment, GAT was utilized for real-time water holdup measurement, and the acquired sensor data were analyzed using three interpolation algorithms: simple linear interpolation, inverse distance weighted interpolation, and Gaussian radial basis function interpolation. The results were subsequently post-processed and visualized with 2020 version MATLAB software, generating two-dimensional representations of water holdup in the wellbore. The study findings demonstrate that, at total flow of 300 m3/d and 500 m3/d, the simple linear interpolation approach yields superior accuracy in water holdup calculations, with imaging outcomes closely aligning with the actual gas-water flow patterns and the authentic gas-water distribution. As total flow and water cut increase, the gas-water two-phase flow progressively shifts from stratified smooth flow to stratified wavy flow. In this paper, the Gaussian radial basis function and inverse distance weighted interpolation algorithms exhibit superior accuracy in water holdup calculations, effectively representing the fluctuating features of the gas-water interface and yielding imaging outcomes that align more closely with experimentally observed gas-water flow patterns.

Eliminating metal artifacts in dental computed tomography using an elaborate sinogram normalization interpolation method with CNR-based metal segmentation

Metal artifact reduction (MAR) in dental cone-beam computed tomography (CBCT) is critical for improving its clinical usefulness and remains a challenging problem. This study presents an elaborate sinogram normalization interpolation (SNI) method with contrast-to-noise ratio (CNR)-based metal segmentation to eliminate metal artifacts in dental CBCT. The proposed MAR method involves three main steps: (1) CNR-based segmentation of a metal trace in the sinogram domain; (2) generation of a residual artifact-reduced prior sinogram; and (3) sinogram completion using the prior sinogram, followed by filtered-backprojection (FBP)-based CBCT reconstruction and metal insertion processes. To verify the efficacy of the proposed method, simulations and experiments were performed using numerical and physical dental phantoms with several metal inserts. The quality of the resulting CBCT images was quantitatively evaluated using the structural similarity (SSIM) metric and compared to those obtained with other interpolation-based MAR methods. A tabletop setup for the micro-CT system was used in the experiment, which comprised an X-ray tube operated under tube conditions of 70 kV p and 5 mA and a flat-panel detector with a pixel size of 198 μm. Our results indicate that the CNR-based metal segmentation method precisely identified traces of metallic objects on the sinogram, and thereby, the proposed SNI-based MAR method considerably reduced metal artifacts in dental CBCT images without introducing any contrast anomalies. The SSIM values of the CBCT images obtained with the proposed MAR method were 0.99 and 0.95 for the simulation and experiment, respectively, which are approximately 11% and 9% greater than those with a simple threshold-based linear interpolation method, respectively, improving the image quality. The proposed MAR method can be applied to reduce metal artifacts in real-world dental CBCT systems.

About Air Quality Assessment using Time Series Analysis Techniques on Air Particles Data

This study investigates the dynamics of air pollution caused by particulate matter (PM10 and PM2.5) in Omsk, Russia. Data collected from the Sensor.Community network between 2021 and 2023 were analyzed using time series analysis methods. The findings reveal a subtle weekly seasonality in the data, potentially attributed to fluctuations in industrial activity and traffic patterns throughout the week. Additionally, a concerning trend of deteriorating air quality was identified, characterized by increasing annual average concentrations of PM10 and PM2.5. The study employed various time series analysis techniques, including Empirical Cumulative Distribution Function (ECDF), Seasonal and Trend decomposition using Loess (STL), and autocorrelation function (ACF) analysis. These methods allowed for the visualization and examination of data distribution, trend identification, and seasonality detection. Furthermore, the Mann-Kendall test confirmed a statistically significant trend of increasing pollution levels over the three-year period. To enhance the accuracy of spatial analysis, data interpolation was performed using kriging with a Gaussian model. This method, compared to simple interpolation, provides a more precise assessment of the spatial distribution of pollutants. The study also implemented Long Short-Term Memory (LSTM) for forecasting hourly pollution values. LSTM exhibited promising results, demonstrating its potential for developing early warning systems, particularly when combined with meteorological data such as wind forecasts. The results highlight the need for effective measures to mitigate air pollution in Omsk and emphasize the importance of utilizing data from diverse sources, including official air quality monitoring stations, to ensure accuracy and reliability in air quality assessments.

A novel direct interpolation boundary element method formulation for solving diffusive–advective problems

The capability of dealing with the advective transport term constitutes a challenging issue for the performance of the majority of the numerical techniques, which significantly lose their precision with the increasing in the relative magnitude of this term. Recently, a new boundary element technique called direct interpolation (DIBEM) has emerged, mainly characterized by the approximation of the entire kernel of the remaining domain integrals. The DIBEM model has been successfully applied to several scalar field problems and recently applied to certain diffusive–advective problems with superior accuracy compared to dual reciprocity formulation (DRBEM). Using DIBEM, a broader range of precise responses for flows with higher Peclet numbers can be reached beyond a lower computational cost due to the simplicity of its matrix operations. These advantages are due DIBEM concept that employs a simple interpolation procedure to approximate the kernel of the advective domain integral. However, it was noticed that in some physical scenarios with spatial variation in the velocity field, the accuracy of DIBEM did not have the same quality observed in other applications. Therefore, this work presents a new DIBEM model so that the integral equations include the explicit calculation of spatial derivatives and simultaneously solve the variation of the divergent velocity field if this is not zero. Conversely, reactive and source terms were also included to expand numerical comparisons. Thus, in the proposed examples, the results of two DIBEM models, the standard (DIBEM-S) and the new, so-called alternative model (DIBEM-A), are compared with DRBEM and also with available analytical solutions.

Influence of Preprocessing Methods of Automated Milking Systems Data on Prediction of Mastitis with Machine Learning Models

Missing data and class imbalance hinder the accurate prediction of rare events such as dairy mastitis. Resampling and imputation are employed to handle these problems. These methods are often used arbitrarily, despite their profound impact on prediction due to changes caused to the data structure. We hypothesize that their use affects the performance of ML models fitted to automated milking systems (AMSs) data for mastitis prediction. We compare three imputations—simple imputer (SI), multiple imputer (MICE) and linear interpolation (LI)—and three resampling techniques: Synthetic Minority Oversampling Technique (SMOTE), Support Vector Machine SMOTE (SVMSMOTE) and SMOTE with Edited Nearest Neighbors (SMOTEEN). The classifiers were logistic regression (LR), multilayer perceptron (MLP), decision tree (DT) and random forest (RF). We evaluated them with various metrics and compared models with the kappa score. A complete case analysis fitted the RF (0.78) better than other models, for which SI performed best. The DT, RF, and MLP performed better with SVMSMOTE. The RF, DT and MLP had the overall best performance, contributed by imputation or resampling (SMOTE and SVMSMOTE). We recommend carefully selecting resampling and imputation techniques and comparing them with complete cases before deciding on the preprocessing approach used to test AMS data with ML models.

Estimating high-resolution soil moisture by combining data from a sparse network of soil moisture sensors and remotely sensed MODIS LST information

ABSTRACT The present work demonstrates a methodology for acquiring high-resolution soil moisture information, namely at 1 km at a daily time step, utilizing data from a sparse network of soil moisture sensors and remotely sensed Land Surface Temperature (LST). Building on previous research and highlighting the strong correlation between surface soil moisture and LST, as a result of the thermal inertia, we first evaluated the correlation between Moderate Resolution Imaging Spectroradiometer (MODIS) LST and ground-based soil moisture information from soil moisture sensors installed in a pilot area in Northeastern Greece. Second, a regression formula was developed for three out of six soil moisture sensors, keeping the three remaining monitoring stations serving as a validation set. Furthermore, regression coefficients were interpolated at 1 km and the regression equations were applied for the entire study area, thus acquiring soil moisture information at a spatial resolution of 1 km at the daily time step. The verification process indicated a reasonable accuracy, with a mean absolute error (MAE) of <0.02 m3/m3. The results were considerably better than using a simple interpolation or downscaled daily 1-km SMAP soil moisture.

Near‐surface wind profiles from numerical model predictions. Part I: Algorithms and comparisons with wind profile based on Monin–Obukhov similarity theory

AbstractWinds are predicted on the discrete grid of numerical weather and climate models. Winds distribute nonlinearly on the height in the near‐surface layer, and a 10 m wind prediction within the layer is often diagnosed upon the Monin–Obukhov similarity theory flux–profile relationship determined from winds at the lowest grid level, the near‐surface atmospheric stability, and surface properties, which leads to concerns that systemic biases may be introduced to the diagnosed wind. Algorithms are proposed to derive near‐surface wind profiles from the grid‐based numerical model forecasts at multiple model levels under the framework of momentum conservations with an implicit solution, associated with simple logarithmic plus linear interpolation in exceptional exemptional conditions. The diagnosed wind profile coheres to the model prediction at the grid level and exhibits differences from the profile using the conventional scheme in the quasi‐steady thermal stratification and non‐steady transitional conditions, retreating to the same logarithmic profile in the neutral condition.

Hydrogen, Oxygen, and Lead Adsorbates on Al13Co4(100): Accurate Potential Energy Surfaces at Low Computational Cost by Machine Learning and DFT-Based Data.

Intermetallic compounds are promising materials in numerous fields, especially those involving surface interactions, such as catalysis. A key factor to investigate their surface properties lies in adsorption energy maps, typically built using first-principles approaches. However, exploring the adsorption energy landscapes of intermetallic compounds can be cumbersome, usually requiring huge computational resources. In this work, we propose an efficient method to predict adsorption energies, based on a Machine Learning (ML) scheme fed by a few Density Functional Theory (DFT) estimates performed on n sites selected through the Farthest Point Sampling (FPS) process. We detail its application on the Al13Co4(100) quasicrystalline approximant surface for several atomic adsorbates (H, O, and Pb). On this specific example, our approach is shown to outperform both simple interpolation strategies and the recent ML force field MACE [arXiv.2206.07697], especially when the number n is small, i.e., below 36 sites. The ground-truth DFT adsorption energies are much more correlated with the predicted FPS-ML estimates (Pearson R-factor of 0.71, 0.73, and 0.90 for H, O and Pb, respectively, when n = 36) than with interpolation-based or MACE-ML ones (Pearson R-factors of 0.43, 0.39, and 0.56 for H, O, and Pb, in the former case and 0.22, 0.35, and 0.63 in the latter case). The unbiased root-mean-square error (ubRMSE) is lower for FPS-ML than for interpolation-based and MACE-ML predictions (0.15, 0.17, and 0.17 eV, respectively, for hydrogen and 0.17, 0.25, and 0.22 eV for lead), except for oxygen (0.55, 0.47, and 0.46 eV) due to large surface relaxations in this case. We believe that these findings and the corresponding methodology can be extended to a wide range of systems, which will motivate the discovery of novel functional materials.

Post-earthquake rapid seismic demand estimation at unmonitored locations via Bayesian networks

Post-earthquake safety assessment of buildings and infrastructure poses significant challenges, often relying on time-consuming visual inspections. To expedite this process, safety criteria based on a demand-capacity model are utilized. However, rapid assessment frameworks require accurate estimations of intensity measures (IMs) to estimate seismic demand and assess structural health. Unfortunately, post-earthquake IM values are typically only available at monitored locations equipped with sensors or monitoring systems, limiting broader assessments. Simple spatial interpolation methods, while possible, struggle to consider crucial physical factors such as earthquake magnitude, epicentral distance, and soil type, leading to substantial estimation errors, especially in areas with insufficient or non-uniform seismic station coverage. To address these issues, a novel framework, BN-GMPE, combining a Bayesian network (BN) and a ground motion prediction equation (GMPE), is proposed. BN-GMPE enables inference and prediction under uncertainty, incorporating physical parameters in seismic wave propagation. A further novelty introduced in this work regards separating the near and far seismic fields in the updating process to attain a clearer understanding of uncertainty and more accurate IM estimation. In the proposed approach, a GMPE is employed for the estimation, and the bias and standard deviation of the prediction error are updated after any new information is entered into the network. The proposed method is benchmarked against a classic Kriging interpolator technique, considering some recent earthquake shocks in Italy. The proposed BN framework can naturally extend for estimating the probability of failure of various structures in a targeted region, which represents the ultimate aim of this research.

Using deep learning to integrate paleoclimate and global biogeochemistry over the Phanerozoic Eon

Abstract. Databases of 3D paleoclimate model simulations are increasingly used within global biogeochemical models for the Phanerozoic Eon. This improves the accuracy of the surface processes within the biogeochemical models, but the approach is limited by the availability of large numbers of paleoclimate simulations at different pCO2 levels and for different continental configurations. In this paper we apply the Frame Interpolation for Large Motion (FILM) deep learning method to a set of Phanerozoic paleoclimate model simulations to upscale their time resolution from one model run every ∼25 million years to one model run every 1 million years (Myr). Testing the method on a 5 Myr time-resolution set of continental configurations and paleoclimates confirms the accuracy of our approach when reconstructing intermediate frames from configurations separated by up to 40 Myr. We then apply the method to upscale the paleoclimate data structure in the SCION climate-biogeochemical model. The interpolated surface temperature and runoff are reasonable and present a logical progression between the original key frames. When updated to use the high-time-resolution climate data structure, the SCION model predicts climate shifts that were not present in the original model outputs due to its previous use of widely spaced datasets and simple linear interpolation. We conclude that a time resolution of ∼10 Myr in Phanerozoic paleoclimate simulations is likely sufficient for investigating the long-term carbon cycle and that deep learning methods may be critical in attaining this time resolution at reasonable computational expense, as well as for developing new fully continuous methods in which 3D continental processes are able to translate over a moving continental surface in deep time. However, the efficacy of deep learning methods in interpolating runoff data, compared to that of paleogeography and temperature, is diminished by the heterogeneous distribution of runoff. Consequently, interpolated climates must be confirmed by running a paleoclimate model if scientific conclusions are to be based directly on them.

Machine learning for gap‐filling in greenhouse gas emissions databases

AbstractGreenhouse gas (GHG) emissions datasets are often incomplete due to inconsistent reporting and poor transparency. Filling the gaps in these datasets allows for more accurate targeting of strategies aiming to accelerate the reduction of GHG emissions. This study evaluates the potential of machine learning methods to automate the completion of GHG datasets. We use three datasets of increasing complexity with 18 different gap‐filling methods and provide a guide to which methods are useful in which circumstances. If few dataset features are available, or the gap consists only of a missing time step in a record, then simple interpolation is often the most accurate method and complex models should be avoided. However, if more features are available and the gap involves non‐reporting emitters, then machine learning methods can be more accurate than simple extrapolation. Furthermore, the secondary output of feature importance from complex models allows for data collection prioritization to accelerate the improvement of datasets. Graph‐based methods are particularly scalable due to the ease of updating predictions given new data and incorporating multimodal data sources. This study can serve as a guide to the community upon which to base ever more integrated frameworks for automated detailed GHG emissions estimations, and implementation guidance is available at https://hackmd.io/@luke‐scot/ML‐for‐GHG‐database‐completion and https://doi.org/10.5281/zenodo.10463104. This article met the requirements for a gold‐gold JIE data openness badge described at http://jie.click/badges.

Efficient estimation procedure for failure probability function by an augmented directional sampling

AbstractFailure probability function (FPF) can reflect quantitative effects of random input distribution parameter (DP) on failure probability, and it is significant for decoupling reliability‐based design optimization (RBDO). But the FPF estimation is time‐consuming since it generally requires repeated reliability analyses at different DPs. For efficiently estimating FPF, an augmented directional sampling (A‐DS) is proposed in this paper. By using the property that the limit state surface (LSS) in physical input space is independent of DP, the A‐DS establishes transformation of LSS samples in standard normal spaces corresponding to different DPs. By the established transformation in different standard normal spaces, the LSS samples obtained by DS at a given DP can be transformed to those at other DPs. After simple interpolation post‐processing on those transformed samples, the failure probability at other DPs can be estimated by DS simultaneously. The main novelty of A‐DS is that a strategy of sharing DS samples is designed for estimating the failure probability at different DPs. The A‐DS avoids repeated reliability analyses and inherits merit of DS suitable for solving problems with multiple failure modes and small failure probability. Compared with other FPF estimation methods, the examples sufficiently verify the accuracy and efficiency of A‐DS.

FlowFit3: Efficient Data Assimilation Of LPT Measurements

Lagrangian particle tracking (LPT) techniques such as Shake-The-Box provide accurate flow measurements by following the movement of tracer particles within a flow. The technique gives measured values of position, velocity and acceleration at the locations of the scattered particles as opposed to a Eulerian grid like with window based techniques such as PIV. This is advantageous for e.g spatially highly resolved flow statistics. Often it is also desirable to have access to a grid-structured representation of the data, especially for the calculation of spatial gradients of flow field measures. While a simple interpolation onto a grid is possible, data assimilation techniques using physics based regularization such as FlowFit2 or VIC+ and VIC#are frequently used nowadays. Through knowledge of the Navier-Stokes equations a higher spatial resolution can be obtained than from interpolation alone. We present the new and highly efficient FlowFit3 approach and its evaluation on synthetic data.

BEM performance with the addition of polynomials solving stationary thermal problems with functional gradation

This work presents the formulation and numerical results of the Dual Reciprocity Boundary Element Method (DRBEM), adapted to include the procedure known as the addition of polynomial functions. This technique is used to improve the interpolation scheme’s accuracy using radial basis functions. This idea works quite effectively in simple interpolation procedures, especially when the field profile to be approximated has similarities with added polynomials. Although applied as a supplementary resource in some works, the efficiency and restrictions of the procedure were not discussed in detail as necessary. In this article, the addition of functions is applied to solve non-homogeneous stationary heat transfer problems. The main objective is to analyze its capability, especially in reducing the demand for internal interpolation points. The three simulations show that although introducing polynomials can improve the results, other characteristics are mandatory for achieving suitable accuracy. Three examples of numerical simulation are solved and show that its effectivity is relative, mainly limited by the type of application.

Real-GDSR: Real-World Guided DSM Super-Resolution via Edge-Enhancing Residual Network

Abstract. A low-resolution digital surface model (DSM) features distinctive attributes impacted by noise, sensor limitations and data acquisition conditions, which failed to be replicated using simple interpolation methods like bicubic. This causes super-resolution models trained on synthetic data does not perform effectively on real ones. Training a model on real low and high resolution DSMs pairs is also a challenge because of the lack of information. On the other hand, the existence of other imaging modalities of the same scene can be used to enrich the information needed for large-scale super-resolution. In this work, we introduce a novel methodology to address the intricacies of real-world DSM super-resolution, named REAL-GDSR, breaking down this ill-posed problem into two steps. The first step involves the utilization of a residual local refinement network. This strategic approach departs from conventional methods that trained to directly predict height values instead of the differences (residuals) and utilize large receptive fields in their networks. The second step introduces a diffusion-based technique that enhances the results on a global scale, with a primary focus on smoothing and edge preservation. Our experiments underscore the effectiveness of the proposed method. We conduct a comprehensive evaluation, comparing it to recent state-of-the-art techniques in the domain of real-world DSM super-resolution (SR). Our approach consistently outperforms these existing methods, as evidenced through qualitative and quantitative assessments.

Reconstruction of High-Resolution 3D GPR Data from 2D Profiles: A Multiple-Point Statistical Approach

Ground-penetrating radar (GPR) is a popular geophysical tool for mapping the underground. High-resolution 3D GPR data carry a large amount of information and can greatly help to interpret complex subsurface geometries. However, such data require a dense collection along closely spaced parallel survey lines, which is time consuming and costly. In many cases, for the sake of efficiency, a choice is made during 3D acquisitions to use a larger spacing between the profile lines, resulting in a dense measurement spacing along the lines but a much coarser one in the across-line direction. Simple interpolation methods are then commonly used to increase the sampling before interpretation, which can work well when the subsurface structures are already well sampled in the across-line direction but can distort such structures when this is not the case. In this work, we address the latter problem using a novel multiple-point geostatistical (MPS) simulation methodology. For a considered 3D GPR dataset with reduced sampling in the across-line direction, we attempt to reconstruct a more densely spaced, high-resolution dataset using a series of 2D conditional stochastic simulations in both the along-line and across-line directions. For these simulations, the existing profile data serve as training images from which complex spatial patterns are quantified and reproduced. To reduce discontinuities in the generated 3D spatial structures caused by independent 2D simulations, the target profile being simulated is chosen randomly, and simulations in the along-line and across-line directions are performed alternately. We show the successful application of our approach to 100 MHz synthetic and 200 MHz field GPR data under multiple decimation scenarios where survey lines are regularly deleted from a dense 3D reference dataset, and the corresponding reconstructions are compared with the original data.

Accurate Trace-Cut and Phase Alignment of Active Ocean-Bottom Seismometer Data

Abstract Accurate positions of ocean-bottom seismometers (OBSs) on the seafloor are critical parameters and can only be obtained by inversion modeling of first-arrival travel times of overhead cross-line airgun shootings. With an increased sampling interval of ≤20 ms for long-term earthquake studies, apparent artifacts affect the phase alignment of first arrivals on the seismic sections of trace-cut airgun shots. Our analysis shows that these apparent misalignments are caused by timing inconsistencies and inaccuracies during the trace-cut, which are so-called rounding errors. To eliminate these rounding errors, a simple interpolation is used to resample traces. Further analysis shows the simple interpolation satisfactorily retains the original waveform. The improved timing accuracy significantly reduces the uncertainty of seafloor locations as shown by Hadal OBS data.

Artificial intelligence warm-start approach: optimizing the generalization capability of QAOA in complex energy landscapes

To address the issue of the quantum approximate optimization algorithm frequently encountering local minima and the cost of parameter optimization within complex non-convex optimization energy landscapes, we consider a warm-start method. This approach leverages the characteristics of transition states in the enhanced optimizer, specifically descending along unique negative curvature directions, to find smaller local minima. Our research results indicate that with the assistance of an enhanced pre-training structure of the AlphaZero AI model, the initialization generalization ability of the new optimizer is significantly enhanced across various test sets. We train on 2-SAT training sets with clause densities between α ≈ 2.6 and α ≈ 2.89, and transfer to more complex test sets. Additionally, the average residual energy density in transfer learning consistently remains below 0.01, even achieving a high transfer success probability of 98% in hard instances with α ≈ 3.7. The search efficiency, pre-trained by ensemble learning, was significantly enhanced, while only requiring simple interpolation of a few transition points to transfer on the global optimal solutions at higher sample clause densities.

Superconvergence error analysis of linearized semi-implicit bilinear-constant SAV finite element method for the time-dependent Navier–Stokes equations

In this paper, based on the scalar auxiliary variable (SAV) approach, the superconvergence error analysis is investigated for the time-dependent Navier–Stokes equations. In which, an equivalent system of the Navier–Stokes equations with three variables and a fully-discrete scheme is developed with semi-implicit Euler discretization for the temporal direction and low-order bilinear-constant finite element discretization for the spatial direction, respectively. With the help of the high-precision estimations of the bilinear-constant finite element pair on the rectangular meshes, the superclose error estimates for velocity in H1-norm and pressure in L2-norm are obtained by treating the trilinear term carefully and skillfully. The global superconvergence results are also derived in terms of a simple and efficient interpolation post-processing technique. Finally, some numerical results are provided to demonstrate the correctness of the theoretical analysis.